Step 1: Preheat your oven to 400°F (200°C). Line a 9-inch springform pan with two sheets of parchment paper, allowing them to extend at least 2 inches above the rim. Crinkle the parchment paper slightly before fitting it into the pan for a rustic look.
Step 2: In a large mixing bowl, beat the softened cream cheese and 1½ cups granulated sugar together using a stand mixer with a paddle attachment or an electric hand mixer until very smooth and creamy, with no lumps. Scrape down the sides of the bowl frequently.
Step 3: Add the eggs one at a time, mixing well after each addition until just combined. Do not overmix. Then, add the egg yolks and vanilla extract, mixing until just incorporated.
Step 4: Gradually pour in the heavy cream, mixing on low speed until the batter is smooth and uniform. Again, avoid overmixing to prevent incorporating too much air.
Step 5: Sift the all-purpose flour into the batter and mix on the lowest speed until just combined and no streaks of flour remain. Be careful not to overmix.
Step 6: Pour the cheesecake batter into the prepared springform pan. Gently tap the pan on the counter a few times to release any large air bubbles.
Step 7: Bake for 60-70 minutes, or until the top is deeply caramelized (almost burnt-looking) and the cheesecake is very jiggly in the center when gently shaken. The edges should be set.
Step 8: Remove the cheesecake from the oven and let it cool completely at room temperature for at least 3-4 hours. Once cooled, transfer it to the refrigerator and chill for at least 6 hours, or preferably overnight, to allow it to fully set.
Step 9: While the cheesecake chills, prepare the cherry compote. In a medium saucepan, combine the pitted cherries, ½ cup granulated sugar, lemon juice, and water. Bring to a simmer over medium heat, stirring occasionally.
Step 10: Continue to simmer the compote for 10-15 minutes, or until the cherries have softened and the sauce has thickened slightly. Remove from heat and stir in the lemon zest, if using. Let the compote cool completely before serving.
Step 11: Once the Basque Burnt Cheesecake is thoroughly chilled, carefully release the springform pan sides and peel away the parchment paper. Slice and serve with generous spoonfuls of the homemade cherry compote.