Preheat your oven to 425°F (218°C). Prepare a 12-cup muffin pan by spraying it with nonstick spray or lining it with cupcake liners. This recipe yields 12 to 14 muffins, so you might need a second pan or plan to bake in batches.
Start by making the crumb topping: In a small bowl, combine the brown sugar, cinnamon, and melted butter. Stir together until well mixed. Add the flour, then gently toss with a fork until the mixture forms crumbly clumps. Be careful not to overmix into a paste. Set the crumb topping aside.
Next, prepare the pumpkin batter: In a large bowl, whisk together the flour, baking powder, baking soda, salt, cinnamon, and pumpkin pie spice. In a separate medium bowl, whisk the brown sugar, granulated sugar, eggs, pumpkin puree, oil, milk, and vanilla extract until smooth. Pour the wet ingredients into the dry ingredients and stir or whisk just until combined. Avoid over-mixing to keep the muffins tender.
For the cheesecake filling: Using a hand mixer or stand mixer with a paddle or whisk attachment, beat the softened cream cheese in a medium bowl on medium-high speed until smooth and creamy. Add the egg yolk, vanilla extract, and sugar, then continue beating until fully combined.
To assemble, spoon about 1 heaping tablespoon of the pumpkin batter into each muffin cup. Add a spoonful (slightly less than a tablespoon) of the cream cheese filling on top of the batter, then cover with another heaping tablespoon of pumpkin batter, or enough to fill the muffin cups completely. Some of the cream cheese filling may peek out from the top or sides—that’s okay! Finally, sprinkle the crumb topping evenly over each muffin and gently press it down to help it stick during baking.
Bake the muffins at 425°F (218°C) for 5 minutes. Then, without removing the muffins from the oven, reduce the heat to 350°F (177°C) and bake for an additional 15 to 17 minutes. The total baking time should be around 20 to 22 minutes. Allow the muffins to cool in the pan for at least 10 minutes before serving.
Store any leftovers tightly covered at room temperature for up to 2 days, or refrigerate for up to one week.