Preheat the oven to 425°F. Line a 12-cup muffin pan with paper liners.

In a small bowl, stir together the flour, brown sugar, cinnamon, and salt for the crumble. Add the melted butter and mix until crumbs form. Refrigerate while you prepare the muffins.

In a medium bowl, beat the cream cheese until smooth. Add the egg, sugar, and vanilla and beat until combined. Set aside.

In a medium bowl, whisk together the flour, baking powder, baking soda, salt, cinnamon, ginger, and nutmeg.

In a large bowl, whisk together the vegetable oil, eggs, sour cream, brown sugar, granulated sugar, and vanilla. Stir in the carrots and walnuts.

Add the dry ingredients to the wet ingredients and fold until just combined.

Divide half of the batter between the prepared muffin cups. Spoon the cream cheese filling into the center of each muffin, then top with the remaining batter. Sprinkle the chilled crumble evenly over the tops.

Bake for 5 minutes. Without opening the oven, reduce the temperature to 350°F and bake for another 18–20 minutes, or until the muffins are set and a toothpick inserted into the muffin portion comes out clean or with a few moist crumbs.

Let cool in the pan for 10 minutes before transferring to a wire rack.