Preheat oven to 375 degrees F (190 degrees C). Grease 12 muffin cups or line with paper liners.
Whisk flour, bran, 3/4 cup brown sugar, baking powder, baking soda, 1 teaspoon cinnamon, and salt in a large bowl until thoroughly combined. Beat milk, vegetable oil, and eggs in a separate bowl.
Pour liquid ingredients into the flour mixture and stir just until mixed. Gently stir apples into the batter and fill prepared muffin cups about 3/4 full.
Mix 1/4 cup flour and 1/4 cup brown sugar with 1/2 teaspoon cinnamon in a bowl; cut the butter into the brown sugar mixture until the streusel topping is crumbly.
Sprinkle about 1 heaping teaspoon of topping onto each muffin.
Bake muffins in the preheated oven until browned and a toothpick inserted into the center of a muffin comes out clean, 25 to 30 minutes.