Step 1: Preheat your oven to 400°F (200°C). Line a 9-inch springform pan with two pieces of parchment paper, crisscrossing them and allowing them to extend several inches above the rim. Crinkle the parchment paper before placing it in the pan to make it more pliable.
Step 2: Prepare the Cherry Compote: In a small saucepan, combine cherries, ¼ cup sugar, and lemon juice. Cook over medium heat, stirring occasionally, until sugar dissolves and cherries soften (5-7 minutes). In a small bowl, whisk cornstarch and cold water until smooth. Stir into the cherry mixture. Cook for another 1-2 minutes until thickened. Remove from heat and let cool completely.
Step 3: Make the Cheesecake Batter: In a large mixing bowl, beat the softened cream cheese with 1½ cups granulated sugar using an electric mixer until smooth and creamy. Scrape down the sides of the bowl.
Step 4: Add the Eggs: Add the eggs one at a time, beating well after each addition until just incorporated. Do not overmix.
Step 5: Incorporate Wet Ingredients: Stir in the heavy cream, vanilla extract, and sea salt until the batter is smooth and combined. Be careful not to overmix the batter to avoid incorporating too much air.
Step 6: Bake the Cheesecake: Pour the batter into the prepared springform pan. Place the pan on a baking sheet. Bake for 50-60 minutes, or until the top is deeply browned (almost black in spots) and the edges are set, but the center is still very jiggly. This burnt look is intentional and desired for a Basque Burnt Cheesecake.
Step 7: Cool and Chill: Turn off the oven and leave the cheesecake inside with the door ajar for 15-20 minutes. Remove from oven and cool completely at room temperature for at least 3-4 hours. The cheesecake will deflate as it cools. Once cool, transfer to the refrigerator and chill for a minimum of 6 hours, or preferably overnight, before serving. Serve slices at room temperature, topped generously with the cooled cherry compote.