Preheat oven to 325°F (160°C).

In a 9-inch microwave-safe pie dish, melt 8 tablespoons of butter. Sprinkle graham cracker crumbs over the melted butter and stir with a fork. Press the mixture evenly into the bottom and up the sides of the pie dish to form the crust. Freeze for at least 30 minutes, or until firm.

In a large bowl, beat softened cream cheese with a hand mixer or stand mixer until fluffy and smooth. Add 0.5 cup of sugar, sour cream, and vanilla extract, and beat until smooth. Add the eggs and mix well until no lumps remain. It's important to use room temperature ingredients for a smooth cheesecake.

Pour the cheesecake filling into the prepared crust.

In a small bowl, melt the remaining 2 tablespoons of butter. Stir in 1 teaspoon of sugar and cocoa powder until combined. Drop spoonfuls of this chocolate mixture in rows across the cheesecake filling. Use a toothpick or the back of a butter knife to create the swirl pattern by dragging it horizontally and then vertically through the chocolate.

Bake for 30-40 minutes, or until the edges of the cheesecake are set and slightly golden brown, and the center has a slight wobble. Cool completely to room temperature, then cover with plastic wrap and chill in the refrigerator for at least 5 hours, or preferably overnight, before serving.