Activate Yeast: In a small bowl, combine water and yeast; let sit until foamy, about 5 minutes.
Mix Dry Ingredients: In a large bowl, whisk together flour, confectioners' sugar, salt, and nutmeg.
Combine Mixtures: Add egg yolks, 2 tablespoons oil, and vanilla to the yeast mixture; whisk until combined. Pour into the flour mixture and stir with a rubber spatula until a slightly sticky dough forms. Cover with plastic wrap and let rise until doubled, 1 to 2 hours.
Prepare to Cut: Line one baking sheet with paper towels and another with parchment, dusted with flour. Dust your countertop and hands with flour. Transfer the dough to the counter, pat into a ¼-inch thick rectangle (about 10 x 12 inches), and cut into 24 two-inch squares. Transfer to the floured baking sheet, spacing them out, and sprinkle lightly with flour.
Fry the Doughnuts: In a large Dutch oven, heat oil to 350°F (or test with a 1-inch cube of bread; it should brown in about 1 minute). Fry 6 dough pieces at a time until golden, about 3 minutes, flipping halfway. Use a slotted spoon to transfer to the paper towel-lined sheet. Repeat with remaining dough.
Fill and Dust: Once cool enough to handle, puncture the side of each doughnut to create a pocket. Fill with 1 to 2 teaspoons of jam or jelly using a piping bag. Dust generously with confectioners’ sugar and serve warm.