Combine the water and yeast in a small bowl and let sit until foamy, about 5 minutes.

Meanwhile, in a large bowl, combine the flour, confectioners' sugar, salt, and nutmeg. Whisk to combine and set aside.

Add the egg yolks, 2 tablespoons of oil, and vanilla to the water/yeast mixture and whisk with a fork until combined.

Add the liquid mixture to the flour mixture and stir with rubber spatula until the dough comes together. It should be a bit sticky. Cover the bowl with plastic wrap (no need to clean it first) and let the dough rise on the countertop until doubled in size, 1 to 2 hours.

Line a baking sheet with a few layers of paper towels. Line another baking sheet with ­parchment paper and dust heavily with flour. Generously dust a clean countertop and your hands with flour. Scrape the dough out of the bowl onto the counter and dust the dough with flour. Pat the dough into ¼-in-thick rectangle (it should be about 10 x 12-inches in size), making sure the bottom doesn't stick and adding more flour to the counter and your hands as needed. Using a pizza wheel or very sharp knife, cut the dough into 24 two-inch squares and transfer to the floured baking sheet, leaving a little space between the squares. Sprinkle the squares lightly with flour.

Add enough of oil to a large Dutch oven or heavy pot to measure about 2 inches deep and heat over medium heat to 350°F. (If you don't have a candy/deep-fry thermometer, drop a 1-in cube of bread in the oil; if it takes about 1 minute to get golden brown, the oil is at the right temperature.) Place 6 dough pieces in the oil and fry until golden brown, about 3 minutes, flipping halfway through frying. Adjust the heat, if necessary, to maintain the oil temperature between 325°F and 350°F. Using a slotted spoon, transfer the donuts to the paper towel-lined baking sheet. Repeat with the remaining donuts.

When the donuts are cool enough to handle, use a paring knife to puncture the side of each to form a pocket in the center. Place the tip of a squeeze bottle or piping bag into the pocket and squeeze 1 to 2 teaspoons of jam or jelly inside. (Alternatively, if you don't have the right tools or just don't want to bother, serve the filling on the side.)

Using a fine sieve, dust the donuts generously with confectioners’ sugar. Serve warm.