Prepare the apple reduction. Add the apple juice or cider to a small saucepan. Simmer over low heat, stirring occasionally, until 1/2 cup remains, about 20 minutes. Pour it into the heat-safe liquid measuring cup used to check the volume, then place it in the refrigerator or freezer while preparing the remaining ingredients. Cool to room temperature before mixing the batter.
Make the crumb topping. Let the melted butter cool slightly. Stir it together with the brown sugar and cinnamon. Add the flour and mix just until large and small crumbs form. Refrigerate while preparing the muffins.
Prepare the oven and pan. Preheat the oven to 425°F. Line one standard 12-cup muffin pan with paper liners.
Combine the dry ingredients. Whisk together the flour, baking powder, baking soda, salt, and apple pie spice.
Combine the wet ingredients. Whisk the oil, brown sugar, and granulated sugar until evenly combined. Whisk in the eggs, followed by the sour cream, vanilla, and cooled apple juice reduction.
Finish the batter. Fold the dry ingredients into the wet ingredients just until no dry flour remains. Fold in the diced apple.
Fill and top the muffins. Use a cookie scoop to divide the batter evenly among the 12 prepared wells; they will be very full. Top each with the crumb topping and press lightly so the crumbs adhere.
Bake the muffins. Bake for 5 minutes at 425°F. Keep the muffins in the closed oven and reduce the setting to 350°F. The initial high heat helps the very full muffins rise quickly, while the lower temperature finishes the centers. Continue baking for 12–14 minutes, checking first at 12 minutes, until the tops are set and a toothpick inserted in the center comes out with a few moist crumbs.
Cool the muffins. Let the muffins cool in the pan for 10 minutes, then transfer them to a wire rack.