Preheat oven to 350 degrees F (175 degrees C). Line a muffin pan with paper liners.

Combine butter, rolled oats, dark brown sugar, 1 tablespoon all-purpose flour, and 1/4 teaspoon cinnamon in a bowl. Mix with a fork or fingers until crumbly.

Whisk 1/2 cup all-purpose flour, white whole-wheat flour, oat flour, baking powder, 1 teaspoon cinnamon, and baking soda together in a large bowl.

Whisk oil, sugar, eggs, and vanilla extract together in a separate bowl. Pour into flour mixture; fold until batter is just combined. Fold peaches and carrots in gently.

Spoon batter into prepared muffin pan, about 2/3 full. Sprinkle with oat topping.

Bake in the preheated oven until an inserted toothpick comes out clean, about 15 minutes. Cool in the pan for 5 to 10 minutes; transfer muffins to a wire rack to cool completely.