Preheat oven to 350° and place oven rack in middle position. Prepare an 8X8 pan or 9 inch pie pan by greasing with butter. Set aside.
Brown butter cobbler
Brown the butter using a pot over low to medium heat, stirring often. This takes about 8-10 minutes. The butter will become fragrant and brown bits will be visible at the bottom.
Scrape the pot of all the brown bits and the melted butter into a bowl. Set aside to cool slightly for 10-15 minutes.
In a separate bowl combine the dry ingredients: sugar, flour, salt and baking powder. Add the vanilla extract to the brown butter and dump into the dry ingredients. Stir to combine using a spatula or wooden spoon. Mixture will be thick and look like cookie dough. Set aside and make the peach filling.
Peach filling
Slice peaches in half and remove the pits. Slice each half into equal, thin slices. Toss together slices with brown sugar and cinnamon. Dump into prepared pan.
Using your fingers crumble the cobbler topping over the peaches. Sprinkle turbinado sugar over the top. Bake in preheated oven at 350° for 30 minutes. After 30 minutes has passed turn the oven temperature up to 450° and bake for another 8-10 minutes. The filling should be bubbling and the top golden brown. It is best served warm, topped with vanilla ice cream, but can be eaten room temperature or even cold.
Notes
While it is best served warm, leftovers can be stored in an airtight container in the refrigerator for 2-3 days. Leftovers can be eaten cold or reheated.