In the bowl of a stand mixer, stir together 3⁄4 cup (180 grams) warm water, 2 tablespoons (24 grams) sugar, and yeast by hand. Let stand until foamy, 5 to 10 minutes.

Add 2¼ cups (281 grams) all-purpose flour, milk, bread flour, melted butter, eggs, vanilla, salt, and remaining 4 tablespoons (48 grams) sugar to yeast mixture; using the paddle attachment, beat at low speed just until combined. Increase mixer speed to medium; beat until well combined, about 1 minute. With mixer on low speed, gradually add remaining 4 cups (500 grams) all-purpose flour, beating just until combined and a shaggy dough forms, about 1 minute.

Switch to the dough hook attachment. Beat at medium-low speed just until dough pulls away from sides of bowl, 8 to 10 minutes. (Dough will still be quite soft and slightly sticky but should not seem excessively wet.)

Spray a large bowl with cooking spray. Place dough in bowl, turning to grease top. Cover and let rise in a warm, draft-free place (75°F/24°C) until doubled in size, about 1 hour.

Line a rimmed baking sheet with plastic wrap. Punch down dough; turn out onto prepared pan. Tightly cover with plastic wrap, and refrigerate for at least 1 hour or up to overnight.

Spray 3 to 4 baking sheets with cooking spray. Cut 15 (5-inch) squares of parchment paper, and place on prepared pans.

Using parchment paper to pick up doughnut holes, transfer doughnut holes in batches to hot oil, letting holes gently fall off parchment into oil. (Do not place parchment in oil.) Fry in batches until golden brown, about 1 minute per side. Using a spider strainer, remove doughnut holes, and let drain on prepared pans.

Using parchment paper to pick up doughnuts, transfer doughnuts in batches to hot oil, letting doughnuts gently fall off parchment into oil. (Do not place parchment paper in oil.) Fry doughnuts in batches until golden brown, about 1 minute per side. Using a spider strainer, remove doughnuts, and let drain on prepared pans. Fill and glaze as desired. Best served same day.