Melt 1/2 tablespoon butter in a saucepan over medium heat. Stir peaches, maple syrup, and a pinch of salt together in the melted butter; cook, stirring occasionally, until the mixture is syrupy, about 10 minutes. Remove from heat and set aside to cool.
Preheat oven to 425 degrees F (220 degrees C). Grease a 10x6-inch baking dish.
Mix 1 cup plus 2 tablespoons flour and 1/2 teaspoon salt together in a bowl. Mash unsalted butter into the flour mixture until lumps are the size of small peas. Blend 2 tablespoons sour cream into the mixture; mix until you have a dough-like mixture. Pat the dough into the bottom and up the sides of the prepared baking dish.
Bake in preheated oven until browned and crust-like, about 10 minutes.
Spread peach mixture over the crust into an even layer.
Beat egg yolks, sugar, 3 1/3 tablespoons sour cream, and 2 tablespoons flour together in a small bowl; pour evenly over the layer of peaches. Cover the dish with aluminum foil.
Bake in the preheated oven for about 35 minutes. Remove aluminum foil and continue baking until a toothpick inserted into the center comes out clean, about 10 minutes. Let cool slightly before serving warm.
