Make the crust: Preheat oven to 350 degrees F. Line the bottom of a 7-inch springform pan with parchment paper. Lightly butter the paper and set aside. Place the crackers, dried cherries, pecans, brown sugar, thyme, paprika, and 1/8 teaspoon of cayenne in a large mixing bowl and toss to combine. Stir in the melted butter and press the mixture into the bottom of the lined pan. Blind bake the crust until set -- about 15 minutes. Cool on a wire rack and set aside.
Make the cheesecake batter: Place the cheeses, sour cream, mustard, salt, and remaining cayenne in a large bowl and mix with an electric mixer set on low speed until well combined. Add the eggs, one at a time, beating well after each addition, until mixture is smooth. Pour the batter into the prebaked crust and bake until set and golden brown -- 40 to 50 minutes. Cool in the pan on a wire rack to room temperature. Remove from pan and serve at room temperature. Cake may be stored, tightly wrapped in plastic and refrigerated, for up to 3 days.