Step 1: Prepare the Donut Dough: In a large bowl, combine the warm milk, yeast, and a pinch of granulated sugar. Let sit for 5-10 minutes until foamy. This indicates the yeast is active.
Step 2: Add the remaining granulated sugar, melted butter, eggs, and vanilla extract to the yeast mixture. Mix well.
Step 3: Gradually add the flour and salt, mixing until a shaggy dough forms. Turn the dough out onto a lightly floured surface and knead for 5-7 minutes until smooth and elastic. The dough should be slightly sticky.
Step 4: Place the dough in a lightly greased bowl, cover with plastic wrap, and let rise in a warm place for 1.5-2 hours, or until doubled in size.
Step 5: Shape the Donuts: Once risen, gently punch down the dough. Roll it out on a lightly floured surface to about ½-inch thickness. Using a 3-inch round cookie cutter (or a glass), cut out donuts. Gather scraps, knead gently, and re-roll to cut more donuts.
Step 6: Place the cut donuts on parchment-lined baking sheets, leaving space between them. Cover loosely with plastic wrap and let rise again for 30-45 minutes, or until visibly puffy.
Step 7: Fry the Donuts: Heat 3-4 inches of vegetable oil in a large heavy-bottomed pot or Dutch oven to 350-360°F (175-180°C).
Step 8: Carefully lower 2-3 donuts into the hot oil (do not overcrowd). Fry for 1.5-2 minutes per side, or until golden brown. Use a slotted spoon to transfer fried donuts to a wire rack lined with paper towels to drain.
Step 9: Prepare the Cheesecake Filling: While donuts cool, beat softened cream cheese and powdered sugar together until smooth. Add milk and vanilla extract, beating until light and fluffy. Transfer to a piping bag fitted with a round tip.
Step 10: Prepare the Peach Cobbler Topping: In a small saucepan, combine peaches, brown sugar, cornstarch, cinnamon, nutmeg, butter, and lemon juice. Cook over medium heat, stirring occasionally, until the mixture thickens and peaches are tender (about 5-7 minutes). Set aside to cool slightly.
Step 11: Assemble the Donuts: Once donuts are cool, insert the piping bag tip into the side of each donut and pipe in about 1-2 tablespoons of cheesecake filling. Top each filled donut with a spoonful of the warm (or room temperature) peach cobbler topping. Serve immediately.