Cook butter in a heavy saucepan over medium heat, stirring constantly until it foams, turns clear, and then turns a deep brown, about 6 minutes.
Pour browned butter into a glass measuring cup and chill in the freezer until solid and chilled, about 1 1/2 hours (Alternatively, brown butter up to a day ahead and let solidify in the refrigerator).
Preheat oven to 400°F and line a baking sheet with a silicone mat or parchment paper. Whisk together flour, sugar, baking powder, baking soda, salt, and cinnamon in a large bowl.
Dice chilled brown butter into 1/2-inch pieces; sprinkle over flour mixture and use a pastry cutter to cut in evenly until mixture resembles coarse meal.
Place peaches between layers of paper towels and press gently to remove moisture; add to flour mixture, folding in gently.
Whisk together buttermilk, egg, and vanilla extract in a small bowl; add to flour mixture and fold in just until moist.
Use a floured bench scraper to scrape the dough out onto a prepared baking sheet; generously flour hands and pat dough into an 8-inch circle.
Score into 12 wedges with floured bench scraper; sprinkle dough with turbinado sugar.
Bake 20 minutes or until golden.
Cool 2 minutes on baking sheet, then carefully transfer to wire rack to cool before slicing into wedges.
Store leftovers in an airtight container up to 1 day and warm before serving