Adjust oven rack to middle position and heat oven to 375 degrees.
Heat butter and sugar in a saucepan until butter melts. Meanwhile, line the bottom of a 9-by-13-inch baking pan with a 12-by-20-inch sheet of foil so it hangs over the long sides of the pan (you'll use this as a 'handle' to pull the cooked dessert from the pan). Arrange graham crackers over foil-lined pan bottom, cutting the final few with a knife so they'll fit snugly.
Pour butter mixture over crackers; spread with knife to cover.
Bake until butter-sugar mixture starts to harden, about 7 minutes.
Remove from oven. Reduce temperature to 300 degrees.
As crust bakes, heat pumpkin, 1 1/4 cups brown sugar and spices in a medium pan over medium-high heat until mixture is sputtery hot. Process eggs on high speed in a blender.
Transfer pumpkin to a pourable container. With blender running, slowly add pumpkin mixture; puree until smooth.
Add cream cheese, one block at a time; puree until smooth.
Pour mixture over crust; bake until set, about 35 minutes.
Meanwhile, mix sour cream, remaining 1/2 cup brown sugar and vanilla.
Remove cheesecake from oven.
Pour sour cream mixture evenly over top; carefully spread so top is completely covered. Return to oven.
Bake to set topping, about 5 minutes longer. Cool to room temperature, then refrigerate (can be made up to 2 days ahead).
To serve, run a knife around the pan perimeter to loosen cake. Use foil handles to pull cake from pan.
