Preheat an oven to 450 degrees F (230 degrees C). Grease a 9-inch round pan.
Mix all-purpose flour, whole-wheat flour, baking powder, sugar, cream of tartar, and salt together in a large bowl. Blend 1/3 cup vegetable oil and butter into the flour mixture until it looks like bread crumbs. Stir milk into the mixture to form a dough. Roll the dough into balls smaller than 1 inch in diameter and arrange into the prepared pan. Drizzle 1/3 cup vegetable oil over the dough.
Sprinkle chopped walnuts over the dough; top with peach cubes. Stir maple syrup and cinnamon together in a small bowl; drizzle evenly over the peaches and walnuts.
Bake in the preheated oven until the edges of the dough is golden brown, 10 to 12 minutes.