Preheat oven to 350°F (175°C). Line a 9x13 inch baking pan with parchment paper, leaving overhang on two sides. Lightly grease the parchment and any exposed edges with nonstick spray.
In a medium bowl, whisk together flour, brown sugar, granulated sugar, cinnamon, and salt. Cut in cold cubed butter with a pastry blender or fork until coarse, pea-sized crumbs form. Chill in the fridge.
In a mixing bowl, beat softened cream cheese until smooth. Add sugar and beat until combined. Add egg and vanilla, mixing just until incorporated. Set aside at room temperature.
In a large bowl, whisk together flour, baking powder, baking soda, salt, pumpkin pie spice, and cinnamon. Set aside.
In another large bowl, whisk together pumpkin puree, granulated sugar, brown sugar, and vegetable oil until smooth. Add eggs one at a time, whisking well after each. Whisk in vanilla and buttermilk until combined.
Add the dry flour mixture to the wet pumpkin mixture in two additions, folding gently with a rubber spatula until just combined. Do not overmix.
Pour half of the pumpkin batter into the prepared pan and spread evenly. Spoon the cream cheese filling over in dollops, then gently spread into an even layer. Top with remaining pumpkin batter, spreading carefully to cover the filling.
Sprinkle the chilled crumb topping evenly over the batter. Bake for 40-45 minutes, until a toothpick inserted into the cake portion comes out mostly clean. The top should be golden brown and the crumbs crisp.
Cool in the pan on a wire rack for at least 30 minutes, then lift out using the parchment overhang. Cool completely before slicing. If using glaze, whisk powdered sugar, milk, and vanilla until smooth and drizzle over the cooled cake.