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1
Place a large, heavy skillet on the stovetop over a medium flame. When hot, add olive oil. Add minced onions and chile pepper. Saute, stirring frequently, until onions begin to release their liquid. Raise heat to high and continue to cook, stirring, until all of the liquid is gone. If onions begin to stick or burn, add small amounts of wine and stir, scraping the bottom. Continue to cook until onions are completely soft and dark brown. Remove from heat and set aside to cool. (Onions will need to be completely cool before they can be added to the cookie dough. To speed up this process, you may put them in a metal container in an ice water bath. Otherwise, you may do this step one day ahead, or just take your time and drink some of that bottle of wine you just opened!)
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2
Place a clean, dry skillet over medium-high heat. Add caraway seeds and shake skillet to evenly distribute them. Leave in skillet, shaking occasionally, until seeds toasted and aromatic, about one to two minutes. Scrape seeds from skillet into a small bowl and set aside.
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3
Using a stand mixer or hand beater, cream butter until light and fluffy. Add caraway seeds, grated cheese, and caramelized onions, and mix well to incorporate. Add egg and beat together. Add flour, one cup at a time, beating until just incorporated. Pat together dough into a ball, wrap with foil or plastic, and stick it in the refrigerator for at least 20 minutes or overnight.
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4
Preheat oven to 350 F. Working in batches, roll the dough out to 1/4-inch thick. Thanks in part to all those onions, the dough can be a touch crumbly and finicky--especially if it chilled overnight, you may need to let it warm up a little bit to be workable. Using a small cookie cutter, cut cookies and place them on a cookie sheet covered with parchment paper or a silicon baking mat. Bake on middle rack until cookies are set and just golden around the edges, about 15 minutes. Let cool slightly on cookie sheet before transferring to a wire rack to cool completely.
