For the crust: Adjust an oven rack to the middle position and heat the oven to 325 degrees. Combine the graham cracker crumbs, melted butter, and sugar together in a medium bowl. Transfer the mixture to a 9-inch springform pan and press evenly into the pan bottom. Bake the crust until fragrant and beginning to brown, 10 to 15 minutes. Let cool on a wire rack while preparing the filling.
For the filling: Raise the oven temperature to 500 degrees. Line a medium bowl with a clean dish towel or several layers of paper towels. Spoon the cottage cheese into the bowl and let drain for 30 minutes.
Process the drained cottage cheese in a food processor until smooth and no visible lumps remain, about 1 minute, scraping down the work bowl once or twice. Add the cream cheese and yogurt cheese and process until smooth, 1 to 2 minutes, scraping down the work bowl once or twice. Add the salt, sugar, lemon zest, and vanilla and continue to process until smooth, about 1 minute, scraping down the work bowl once or twice. With the processor running, add the eggs, one at a time and process until smooth.
Being careful to not disturb the baked crust, spray the sides of the springform pan with vegetable oil spray. Set the springform pan on a rimmed baking sheet. Pour the filling into the cooled crust and bake 10 minutes. Without opening the oven door, reduce the oven temperature to 200 degrees and continue to bake until an instant-read thermometer inserted into the center of the cheesecake reads 150 degrees, about 1 1/2 hours.
Transfer the cake to a wire rack, and run a paring knife around the edge of the cake. Let cool until barely warm, 2 1/2 to 3 hours, running a paring knife around the edge of the cake every hour or so. Wrap the pan tightly in plastic wrap, and refrigerate until cold, at least 3 hours.
To unmold the cheesecake, wrap a hot kitchen towel around the springform pan and let stand for 10 minutes. Remove the sides of the pan and blot any excess moisture from the top of the cheesecake with paper towels. Let the cheesecake stand at room temperature about 30 minutes, then cut into wedges.