Combine the flour, sugar, yeast, salt, milk, butter, eggs, egg yolks, lemon zest, and vanilla in the bowl of a standing mixer fitted with the paddle attachment (the hook is not needed here, since this is a rather loose batter) and mix on low speed until smooth, 8 to 10 minutes.
Transfer the dough to a lightly oiled bowl or dough-rising container, cover the bowl with plastic wrap, and let stand at room temperature until doubled in volume, about 1 hour.
Line a rimmed baking sheet with parchment or waxed paper. Brush a thin layer of vegetable oil on the paper. Turn the dough onto a lightly floured countertop and divide into 16 equal pieces. Shape each piece into a round and then flatten slightly with the palm of your hand. Place the rounds seam side down on the oiled paper. Let stand uncovered at room temperature until doubled in volume, about 1 hour.
Add 3 inches of oil to a deep pot. Attach a candy thermometer to the side of the pot and heat the oil over medium heat until it reaches 350°F. Line a large platter or baking sheet with several layers of paper towels.
Place a few of the doughnuts in the hot oil, being careful not to crowd them in the pot.
Fry until golden brown on one side, about 1 minute. Turn and continue to fry until golden on both sides, another minute or two. Don’t worry that there is a line of light-colored dough around the perimeter of each doughnut. This will be the “soft spot” where you will insert the tip of a pastry bag to pipe in some jam. Drain the cooked doughnuts on the paper towels, roll them in sugar, and repeat with the remaining dough, keeping an eye on the thermometer to make sure the oil stays at a steady 350°F and adjusting the heat as necessary.
To fill the doughnuts, place the jam in a bowl and stir it thoroughly so that there are no lumps. Scrape it into a pastry bag fitted with a large plain tip. Insert the tip into the soft spot of each doughnut and squeeze about 1 tsp. of the jam into the center. Serve immediately.