Filled Doughnuts

Make the dough: Combine water and yeast in a large bowl, stir until yeast dissolves, and let stand for 5 minutes. Stir in 1 cup flour, cover with plastic wrap, and let stand until bubbly -- about 30 minutes. Lightly coat a large bowl with 1 teaspoon butter and set aside.

Place remaining butter in a large bowl and beat in a stand mixer fitted with a paddle attachment or by hand with a wooden spoon until creamy -- about 1 minute. Add sugar and beat until light and fluffy. Add eggs and yolks, one at a time, beating after each addition until well incorporated. Add the vanilla and beat until combined. Reduce mixer speed to low and add the yeast mixture and the salt.

Gradually add the flour and mix until the flour is fully incorporated and the dough begins to pull away from the sides of the bowl. Place in the prepared bowl and turn to coat all sides. Cover the bowl with plastic wrap and let rise in a warm, draft-free place until doubled in volume -- 1 1/2 to 2 hours. Punch down dough, remove from the bowl, and wrap tightly in plastic wrap. Place the wrapped dough in a sealable plastic bag and refrigerate the dough for 3 or up to 12 hours.

Cook the doughnuts: Line 2 baking sheets with parchment paper; sprinkle each lightly with flour and set aside. Turn the dough out onto a generously floured work surface and roll it out to about 1/4 inch thick. Cut out doughnuts using a 3 1/2-inch round cutter and transfer them to the prepared baking sheets. Gather dough scraps, reroll, and repeat cutting doughnuts until all the dough is used. Let rise, uncovered, in a warm draft-free place until puffy -- about 30 minutes.

Fry the doughnuts: Heat about 4 inches of vegetable oil in a large deep skillet fitted with a thermometer over medium-high heat until oil reaches 365° F. Fry the doughnuts three at a time until golden -- about 2 minutes per side. Remove doughnuts with a slotted spoon and drain on a wire rack.

Fill the doughnuts: Place filling in a pastry bag fitted with a long, narrow pastry tip, such as a Bismarck tip. While doughnuts are still slightly warm, insert the tip into the inside of a doughnut. Apply light, continuous pressure to the pastry bag while slowly moving the tip slightly from side to side and pulling the tip out of the doughnut. Serve immediately.