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Stir together yeast and warm milk in mixer bowl and let stand until foamy, about 5 minutes. (If mixture doesn’t foam, start over with new yeast.) Mix in 3/4 cup flour at low speed until combined. Cover bowl with a kitchen towel and let dough rise in a draft-free place at warm room temperature until doubled and bubbles appear on surface, about 1 hour.
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At low speed, mix in egg, 2 Tbsp duck fat, sugar, salt, and remaining 3/4 cup flour until combined, then beat at medium speed until smooth and elastic, 5 to 7 minutes. Scrape dough into center of bowl and dust lightly with additional flour. Cover bowl with kitchen towel and let dough rise at warm room temperature until doubled, about 1 hour.
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Discard skin and bones from duck confit, then shred meat. Stir together meat and preserves.
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Line a baking sheet with parchment or wax paper and lightly dust with flour. Punch down dough (it will be soft) and turn out onto a lightly floured surface. Cut into 16 equal pieces. With lightly floured hands, flatten 1 piece of dough and put a heaping tsp duck confit mixture in center. Gather dough up and around filling and pinch to enclose. Roll into a ball and transfer to baking sheet. Make 15 more balls, arranging 1 inch apart on sheet.
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Heat 2 inches duck fat in a 2-qt heavy saucepan over medium heat to 350°F. Fry doughnuts in batches of 4, turning frequently, until puffed and golden, about 2 minutes per batch. Transfer with a slotted spoon to paper towels to drain. Return oil to 350°F between batches. Serve doughnuts hot, with additional preserves.