Step 1: Preheat your oven to 400 degrees (or set up a grill for indirect grilling and preheat to the same). Lightly oil the springform pan. Wrap the outside with aluminum foil.
Step 2: Make the crust: Break the cookies into pieces and grind to a fine powder with the 2 tablespoons light brown sugar in the food processor. Add the melted butter and pulse the processor to obtain a crumbly dough. Press it into the bottom and halfway up the sides of the springform pan. Bake the crust until lightly browned, 5 to 8 minutes. Transfer to a wire rack and let cool.
Step 3: Make the filling. Wipe out the processor bowl or use a stand mixer. Add the cream cheese, 1 cup sugar, vanilla, and lemon zest and process/beat until smooth. Work in the eggs one by one, beating/processing until smooth after each addition. Pour the batter into the springform pan. Tap the pan on the counter a few times to knock out any air bubbles.
Step 4: Meanwhile, light your smoker according to the manufacturer’s instructions and preheat to 300 degrees. Add the wood as specified by the manufacturer.
Step 5: Smoke the cheesecake until the top is bronzed with smoke and the filling is set, 1-1/2 to 2 hours or more. To test for doneness: gently poke the side of the pan—the filling will jiggle, not ripple. Alternatively, insert a slender metal skewer in the center: it should come out clean. Transfer the cheesecake to a wire rack to cool to room temperature. Refrigerate until serving. (The cheesecake can be made up to 8 hours ahead.) Let the cheesecake warm slightly at room temperature before serving.
Step 6: Run a slender knife around the inside of the springform pan. Loosen and remove the ring. If serving the Burnt Sugar Cream Sauce, pour it over the cheesecake (or serve on the side). Cut into slices and dig in.