In a large bowl, combine the flour, 1/4 cup sugar, salt, and yeast.
In a medium bowl, combine the milk, 2 tablespoons butter, egg, and vanilla extract. Cover with plastic wrap and sit at room temperature for 10 minutes.
Either by hand, or using a stand mixer, combine the dry ingredients with the wet. Mix until the liquid is absorbed and the dough pulls off the walls of the mixing bowl, approximately 5 minutes.
Remove dough from bowl, knead by hand into a ball, and return to bowl. Cover with plastic wrap and proof at room temperature until doubled in size, approximately 2 hours.
Place dough on a clean countertop, and with a rolling pin, roll dough till 1/2-inch in thickness.
Using a 2-inch cookie cutter, cut out 12 round discs. Re-roll the dough if necessary to complete the cuts.
Place discs on a sheet pan, cover with a kitchen towel, and set at room temperature until the dough swells, approximately 45 minutes.
Prepare the grill with a baking stone for indirect cooking over medium heat (350º-450ºF).
Place the discs on the baking stone and grill over indirect medium heat with the lid closed as much as possible until the donuts have risen and are golden brown, approximately 10–12 minutes.
Remove from the grill. Brush with remaining butter and roll in remaining sugar.
Use a chopstick or similar tool to pierce a hole in the side of each donut. Using a piping bag or gallon-sized plastic bag with the corner cut off filled with jelly, squeeze jelly into the hole of each donut.
