Combine yeast, sugar, and 1 cup milk. Let stand until foamy, about 8 minutes.
Whisk flour and salt in the bowl of a standing mixer fitted with the dough hook attachment. Add yeast mixture, eggs, and butter, and beat on medium-low speed until dough is soft but not sticky, about 3 minutes.
On a lightly floured surface, knead dough until smooth and elastic, 3 to 4 minutes. Transfer dough to a medium bowl coated with cooking spray, and cover loosely with plastic wrap. Let rise in a warm, draft-free place until doubled in size, about 1 1/2 hours.
Punch down the dough. On a lightly floured surface, knead dough a few times, and roll out to 1/4 inch thick. Cover with a clean kitchen towel, and let rest for 5 minutes. Using a 2-inch cookie cutter, cut out rounds, and transfer to a lightly floured baking sheet. Reroll scraps, and cut out. Cover rounds with kitchen towel, and let rise slightly in a warm, draft-free place for 20 minutes.
Meanwhile, heat oil in a pot until it reaches 375. Line a wire rack with paper towels. Working in batches of 4 or 5, add doughnuts to oil, and fry, turning once, until golden and puffed, about 1 minute per side. Using a slotted spoon, place doughnuts on rack to cool.
Place a wire rack on parchment or a baking sheet. Whisk together confectioners' sugar and remaining 2 tablespoons milk in a shallow bowl. Spoon jam into a pastry bag fitted with a plain 3/8-inch tip (such as Ateco #804). Pierce a hole in the side of a doughnut with the tip, and squeeze in jam to fill (filled doughnuts will feel heavy). Place on rack. Repeat.
Holding filled doughnuts by their sides, gently dip tops into icing and return to rack. Immediately sprinkle with sanding sugar. Doughnuts are best when served immediately, but they can be stored in airtight containers overnight.