In a small bowl, stir together warm milk, 1 teaspoon (4 grams) granulated sugar, and yeast. Let stand until foamy, about 5 minutes.
In the bowl of a stand mixer, whisk together 3 cups (375 grams) flour, 1 tablespoon (9 grams) salt, and remaining 3 tablespoons plus 2 teaspoons (56 grams) granulated sugar by hand. Add warm milk mixture, eggs, and 2 teaspoons (8 grams) vanilla; using the paddle attachment, beat at low speed until combined, about 1 minute. Slowly add ½ cup (63 grams) flour, beating until a shaggy dough forms; scrape sides of bowl and paddle.
Switch to the dough hook attachment. Beat at medium-low speed until dough becomes smooth, elastic, and slightly tacky, 6 to 8 minutes; add up to remaining ½ cup (63 grams) flour, 1 tablespoon (8 grams) at a time, if dough is too sticky. With mixer on medium-low speed, add butter, 1 tablespoon (14 grams) at a time, beating until combined after each addition (about 8 minutes total). Beat until a smooth, elastic dough forms, 5 to 7 minutes. Turn out dough onto a lightly floured surface; knead dough 5 to 8 times, and shape into a smooth round.
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, 45 minutes to 1 hour.
Line 2 baking sheets with parchment paper.
In a medium bowl, stir together peaches, cinnamon, and remaining ½ teaspoon (1.5 grams) salt.
Punch down dough, and turn out onto a lightly floured surface. Roll dough into an 18×12-inch oval. Place peaches lengthwise down half of oval; fold plain half of dough over peaches. Using a bench scraper or pastry wheel, cut dough diagonally lengthwise into 1-inch-wide strips; repeat in other direction to create diamond-shaped pieces. Press cut dough pieces back together into a 15×4-inch rectangle. (See Note.) Cut rectangle into 10 pieces (about 1½ inches wide each). Press each fritter together to ensure dough has adhered to itself, and place on prepared pans. Cover and let rise in a warm, draft-free place (75°F/24°C) until puffed, about 30 minutes.
In a large Dutch oven, pour oil to a depth of 2 inches, and heat over medium-high heat until a deep-fry thermometer registers 350°F (180°C).
Cut parchment around fritters so each fritter is on a small piece of parchment. Working in batches, carefully transfer fritters on parchment to hot oil. Fry until golden brown, 2 to 3 minutes per side. (Parchment paper will float away from fritter as it cooks; remove with tongs, and discard.) Using a slotted spoon or spider strainer, remove fritters from oil, and let drain on paper towels.
In a small bowl, whisk together confectioners’ sugar, milk, and remaining 1 teaspoon (4 grams) vanilla. Drizzle onto warm fritters. Serve immediately.