Activate Yeast Mixture:
In a medium bowl or liquid measuring cup, combine milk and 1 teaspoon sugar, stirring until sugar dissolves. Scatter yeast over the surface. Let stand until yeast becomes foamy and bubbly, roughly 5 minutes.Mix and Knead Dough:
Coat a large bowl with cooking spray. In another large bowl, combine flour, salt, and 1/4 cup sugar, whisking thoroughly. Create a well in the middle, then add yeast mixture, butter, and vanilla. Crack eggs into the well. Using a wooden or silicone spoon, mix until a shaggy dough forms. Turn dough out onto a lightly floured surface and knead until soft and smooth, approximately 6 minutes. Shape into a tight ball. Transfer to the greased bowl, cover tightly with plastic wrap, and let rest until doubled in volume, 1 hour to 1 hour 30 minutes.Cut and Rise Dough:
Lightly flour 2 baking sheets. On a floured surface, roll dough to 1/2" thickness. Use a 3" round cutter to cut out rounds. Gently gather scraps, reroll, and cut additional rounds without overworking the dough (you should get about 12). Place rounds on prepared sheets, cover with plastic wrap, and let rise until puffy, 30 to 45 minutes.Heat Oil for Frying:
Pour remaining 1/2 cup sugar into a large bowl. Fill a large heavy pot with oil to a depth of 1" and attach a deep-fry thermometer. Heat over medium-high heat until thermometer reads 350°.Fry and Coat Donuts:
Fry rounds in batches, flipping once and adjusting heat to keep oil temperature steady, until golden and puffed, 1 to 2 minutes per side. Move to a wire rack placed over a baking sheet, letting excess oil drain for 15 seconds. Toss in sugar to coat the outside of each donut. Return to rack and cool completely. Repeat with remaining rounds.Fill Donuts with Jelly:
Attach coupler and 1/4" tip to pastry bag. In a medium bowl, whisk jelly until smooth. Fill bag with jelly. Using a chopstick or wooden skewer, poke a hole in the side of each donut. Insert tip and pipe jelly until the center feels slightly puffed and offers some resistance. Arrange on a platter.