SUMMARY:
According to tradition, the episode of the Golden Calf in the Book of Exodus took place on the day that would later become Yom Kippur (the 10th of Tishrei).
Rabbi David Kasher (IKAR in Los Angeles) has recently cited a text [Pachad Yitzchak (R' Yitzchak Hutner, 1906-1980), Purim #21] to the effect that Yom Kippur and Purim are related holidays. This notion comes from the fact that Yom (ha-)Kippurim could be read alternatively as "k-purim" ("like Purim"), as well as a similarity of wording in descriptions of imminent destruction related to Yom Kippur and Purim. (see below)
The participants in Rabbi Kasher's class suggested ways of understanding such a similarity. Obviously, both Yom Kippur and Purim involve fasting. But I've found a fascinating link between the Golden Calf event which took place on Yom Kippur and a particular verse from the Book of Esther (3:15). This link involves a similarity of wording between that Esther verse and the description of the Golden Calf event in Exodus 32:6 .
פחד יצחק, פורים כא
בענינו של יום הכיפורים נאמר ׳׳הרף ממני ואשמידם׳׳
אלא שביום הכיפורים גם הגזרה וגם הישועה לא היתה ניכרת אלא למעלה, בדיבורים שבין קוב׳׳ה למשה.
Pachad Yitzchak (R' Yitzchak Hutner, 1906-1980), Purim #21
On Yom Kippur [the day Moses descended from the mountain], it says, "Let me alone, and I will destroy them..." (Deut. 9:14)
Whereas on Purim, it says, "To destroy, to kill, and to annihilate..." (Esther 9:22)
Because on Yom Kippur, both the decree and the exoneration were only known up above, in the conversation between the Holy One and Moses. Whereas on Purim, both the decree and the exoneration were known down here, in the dealings between Achasverosh, Mordechai and Haman.
In Exodus 32:1-6, the Israelites are alarmed that Moses has not descended according to schedule from his 40-day Mt. Sinai task. Aaron dubiously strives to allay the peoples' fears by inviting them to donate their jewelry for the making of a molten Golden Calf and proclaiming that the next day shall be a "festival of Hashem".
In verse 32:6 (highlighted below), on that next day the people offer sacrifices then sit down sit down to eat and drink and make merry:
(א) וַיַּ֣רְא הָעָ֔ם כִּֽי־בֹשֵׁ֥שׁ מֹשֶׁ֖ה לָרֶ֣דֶת מִן־הָהָ֑ר וַיִּקָּהֵ֨ל הָעָ֜ם עַֽל־אַהֲרֹ֗ן וַיֹּאמְר֤וּ אֵלָיו֙ ק֣וּם ׀ עֲשֵׂה־לָ֣נוּ אֱלֹהִ֗ים אֲשֶׁ֤ר יֵֽלְכוּ֙ לְפָנֵ֔ינוּ כִּי־זֶ֣ה ׀ מֹשֶׁ֣ה הָאִ֗ישׁ אֲשֶׁ֤ר הֶֽעֱלָ֙נוּ֙ מֵאֶ֣רֶץ מִצְרַ֔יִם לֹ֥א יָדַ֖עְנוּ מֶה־הָ֥יָה לֽוֹ׃
(ב) וַיֹּ֤אמֶר אֲלֵהֶם֙ אַהֲרֹ֔ן פָּֽרְקוּ֙ נִזְמֵ֣י הַזָּהָ֔ב אֲשֶׁר֙ בְּאׇזְנֵ֣י נְשֵׁיכֶ֔ם בְּנֵיכֶ֖ם וּבְנֹתֵיכֶ֑ם וְהָבִ֖יאוּ אֵלָֽי׃
(ג) וַיִּתְפָּֽרְקוּ֙ כׇּל־הָעָ֔ם אֶת־נִזְמֵ֥י הַזָּהָ֖ב אֲשֶׁ֣ר בְּאׇזְנֵיהֶ֑ם וַיָּבִ֖יאוּ אֶֽל־אַהֲרֹֽן׃
(ד) וַיִּקַּ֣ח מִיָּדָ֗ם וַיָּ֤צַר אֹתוֹ֙ בַּחֶ֔רֶט וַֽיַּעֲשֵׂ֖הוּ עֵ֣גֶל מַסֵּכָ֑ה וַיֹּ֣אמְר֔וּ אֵ֤לֶּה אֱלֹהֶ֙יךָ֙ יִשְׂרָאֵ֔ל אֲשֶׁ֥ר הֶעֱל֖וּךָ מֵאֶ֥רֶץ מִצְרָֽיִם׃
(ה) וַיַּ֣רְא אַהֲרֹ֔ן וַיִּ֥בֶן מִזְבֵּ֖חַ לְפָנָ֑יו וַיִּקְרָ֤א אַֽהֲרֹן֙ וַיֹּאמַ֔ר חַ֥ג לַיהוה מָחָֽר׃
(ו) וַיַּשְׁכִּ֙ימוּ֙ מִֽמׇּחֳרָ֔ת וַיַּעֲל֣וּ עֹלֹ֔ת וַיַּגִּ֖שׁוּ שְׁלָמִ֑ים
וַיֵּ֤שֶׁב הָעָם֙ לֶֽאֱכֹ֣ל וְשָׁת֔וֹ וַיָּקֻ֖מוּ לְצַחֵֽק׃ {פ}
(1) When the people saw that Moses was so long in coming down from the mountain, the people gathered against Aaron and said to him, “Come, make us a god who shall go before us, for that fellow Moses—the man who brought us from the land of Egypt—we do not know what has happened to him.”
(2) Aaron said to them, “Take off the gold rings that are on the ears of your wives, your sons, and your daughters, and bring them to me.”
(3) And all the people took off the gold rings that were in their ears and brought them to Aaron.
(4) This he took from them and cast in a mold, and made it into a molten calf. And they exclaimed, “This is your god, O Israel, who brought you out of the land of Egypt!”
(5) When Aaron saw this, he built an altar before it; and Aaron announced: “Tomorrow shall be a festival of יהוה !”
(6) Early next day, the people offered up burnt offerings and brought sacrifices of well-being;
they sat down to eat and drink, and then rose to dance (or: "to make merry").
This merry-making can be linked through a similarity of language to certain verses in the Book of Esther (3:13-15) , especially verse 15 (highlighted below). Haman's decree has been published, and while King Ahashverosh and Haman sit down for a feast, the Jewish people are distraught ("dumbfounded") about the situation:
(13) וְנִשְׁל֨וֹחַ סְפָרִ֜ים בְּיַ֣ד הָרָצִים֮ אֶל־כׇּל־מְדִינ֣וֹת הַמֶּ֒לֶךְ֒ לְהַשְׁמִ֡יד לַהֲרֹ֣ג וּלְאַבֵּ֣ד אֶת־כׇּל־הַ֠יְּהוּדִ֠ים מִנַּ֨עַר וְעַד־זָקֵ֜ן טַ֤ף וְנָשִׁים֙ בְּי֣וֹם אֶחָ֔ד בִּשְׁלוֹשָׁ֥ה עָשָׂ֛ר לְחֹ֥דֶשׁ שְׁנֵים־עָשָׂ֖ר הוּא־חֹ֣דֶשׁ אֲדָ֑ר וּשְׁלָלָ֖ם לָבֽוֹז׃
(14) פַּתְשֶׁ֣גֶן הַכְּתָ֗ב לְהִנָּ֤תֵֽן דָּת֙ בְּכׇל־מְדִינָ֣ה וּמְדִינָ֔ה גָּל֖וּי לְכׇל־הָֽעַמִּ֑ים לִהְי֥וֹת עֲתִדִ֖ים לַיּ֥וֹם הַזֶּֽה׃
(15) הָֽרָצִ֞ים יָצְא֤וּ דְחוּפִים֙ בִּדְבַ֣ר הַמֶּ֔לֶךְ וְהַדָּ֥ת נִתְּנָ֖ה בְּשׁוּשַׁ֣ן הַבִּירָ֑ה וְהַמֶּ֤לֶךְ וְהָמָן֙ יָשְׁב֣וּ לִשְׁתּ֔וֹת וְהָעִ֥יר שׁוּשָׁ֖ן נָבֽוֹכָה׃ {ס}
Esther 3:13-15 (especially verse 15)
(13) Accordingly, written instructions were dispatched by couriers to all the king’s provinces to destroy, massacre, and exterminate all the Jews, young and old, children and women, on a single day, on the thirteenth day of the twelfth month—that is, the month of Adar—and to plunder their possessions.
(14) The text of the document was to the effect that a law should be proclaimed in every single province; it was to be publicly displayed to all the peoples, so that they might be ready for that day.
(15) The couriers went out posthaste on the royal mission, and the decree was proclaimed in the fortress Shushan. The king and Haman sat down to feast, but the city of Shushan was dumfounded.
Through this similarity of phrasing in Exodus 32:6 and Esther 3:15, we find an unexpected link between Yom Kippur (the day of the Golden Calf episode) and the holiday of Purim. This serves to augment our understanding of the Pachad Yitzchak material which opened this discussion.