Make the crust: Preheat the oven to 350 degrees F. Butter a 9-by-13-inch baking dish. Line with foil, leaving a 2-inch overhang on the long sides; butter the foil. Combine the cracker crumbs, melted butter and sugar in a bowl. Press evenly into the bottom of the prepared pan. Bake until toasted, about 10 minutes. Transfer to a rack and let cool 15 minutes.
Make the filling: Beat the cream cheese and sugar in a large bowl with a mixer on medium-high speed until smooth, about 1 minute. Add the eggs, sour cream and vanilla and beat until combined, 1 minute. Remove 1 cup of the batter to a medium bowl and stir in the peanut butter until smooth. Pour the peanut butter batter onto the crust in a thin even layer. Pour the remaining cheesecake batter on top; smooth with a spatula.
Bake until the cheesecake is set around the edges but still jiggly in the center, about 30 minutes. Transfer to a rack to cool completely, about 1 hour. Refrigerate until set, at least 2 hours or overnight.
Slide an offset spatula or paring knife around the edge of the pan, then remove the cheesecake using the foil overhang. Slice into squares and top with sprinkles.