Preheat your oven to 350°F (175°C) and grease a 9×9-inch baking pan.
In a large bowl, beat together butter and sugar until light and fluffy. Add eggs, pumpkin purée, milk, and vanilla extract, mixing well.
In another bowl, whisk together flour, pumpkin pie spice, baking powder, baking soda, and salt. Gradually combine the dry mixture with the wet ingredients until smooth.
For the cream cheese layer, beat cream cheese, sugar, egg, and vanilla extract in a medium bowl until smooth.
Pour half of the pumpkin batter into the prepared baking pan. Gently spread the cream cheese mixture over the batter, then top with the remaining pumpkin batter.
In a small bowl, use a fork to combine brown sugar, flour, cold butter, and cinnamon until it forms coarse crumbs. Sprinkle evenly over the top of the cake.
Bake for 40–45 minutes, or until a toothpick inserted into the center comes out clean.
Let the cake cool in the pan before slicing. Serve with coffee or tea for a delightful treat!