Combine the diced peaches, brown sugar, cinnamon, salt, and lemon juice in a saucepan over medium heat. Cook for about 8 to 10 minutes, stirring occasionally, until the peaches soften and release their juices. Stir in the cornstarch slurry and vanilla, then cook for another minute until thickened. Transfer to a bowl and refrigerate until completely cool.
Warm the milk to about 110°F, then stir in the sugar and yeast. Let it sit for 5 to 7 minutes until it turns foamy and fragrant. If it doesn't foam, the yeast may be inactive, so it's worth starting over with a fresh packet. This step confirms your dough will actually rise properly later.
Add the eggs, yogurt, melted butter, flour, and salt to the yeast mixture, then mix on low for about a minute. Let the shaggy dough rest for 5 minutes so the flour can absorb the liquid, then knead for 6 to 8 minutes until smooth. The dough should feel slightly tacky but not sticky to the touch.
Place the dough in a lightly oiled bowl, cover it, and set it somewhere warm for 1 to 1.5 hours. You're looking for it to double in size, which is what gives these fritters their soft, fluffy interior. Don't rush this step even if you're short on time.
Roll the risen dough into a large rectangle, then spread the cooled peach filling down the center, leaving the edges clear. Fold the sides over to enclose the filling, then cut the dough into a rough grid using a bench scraper or knife. Toss and gather the pieces together, cutting a few more times until you have small, filling-flecked chunks.
Gather a handful of dough pieces and press them firmly together into a rough disc about 1/4 to 1/2 inch thick. Don't be shy with extra flour here, since a well-packed fritter is what keeps it from falling apart in the oil. Place each shaped fritter on its own floured parchment square.
Transfer the shaped fritters, still on their parchment squares, to the refrigerator or freezer while your oil heats up. This short chill firms up the dough so the fritters hold their shape once they hit the hot oil. Ten to fifteen minutes is usually enough.
Heat the oil to between 360°F and 375°F, then carefully lower two fritters at a time into the pot using their parchment squares. After about a minute, the paper will release naturally and can be pulled away with tongs. Fry for 2 to 3 minutes per side until deeply golden, then transfer to a wire rack.
While the oil heats, melt the butter in a saucepan over medium heat, swirling often until it turns golden brown and smells nutty. Whisk in the powdered sugar, maple syrup, milk, and vanilla until smooth and glossy. Keep it warm so it stays pourable for dipping.
Dip each warm fritter into the glaze while it's still hot from the fryer, turning it so the glaze coats every crack and crevice. Set it back on the wire rack for a minute to let the glaze set slightly. Serve them warm for the best texture and flavor.