Prep your oven and pan. Position a rack in the center of the oven and preheat to 400°F. Lightly spray a 9-inch springform cake pan with cooking spray and line the bottom with two sheets of parchment paper, making sure that the paper sticks up 2 inches above the sides of the pan. The parchment paper won't lie flat—that's totally okay, I promise!
Beat the cream cheese and sugar. In the bowl of a stand mixer fitted with the paddle attachment, beat the cream cheese on medium until soft and creamy, 1 to 2 minutes. Reduce the mixer to low and add the sugar and beat until light and fluffy, about 1 minute, using a rubber spatula to scrape down the bottom and sides of the bowl as necessary. Add the eggs and the cream. With the mixer on low, add the eggs one at a time, adding the next egg only after the previous one has been fully incorporated, scraping down the bottom and sides of the bowl after each addition. Slowly pour the cream down the side of the bowl and beat until combined, about 1 minute.
Add the flour and salt. With the mixer on low, gradually add the flour and salt and beat until just combined. Scrape down the bottom and sides of the bowl once more, and beat on low for an additional 30 seconds.
Prep for baking. Pour the batter into the prepared pan. Pick up the pan and whack it a few times on your kitchen counter to allow the batter to fill the the pan more evenly.
Bake the cheesecake. Bake for 50 to 55 minutes, or until the edges of the cheesecake are set but the center still wobbles slightly. Cool completely in its pan on a wire rack before slicing.
Serve and store. To serve, unlatch the springform pan and lift the sides of the pan away from the bottom of the pan. Use a hot knife to slice and serve. Serve slightly warm, at room temperature, or chilled from the refrigerator. The burnt basque cheesecake can be kept at room temperature, under a cake dome or a large bowl turned upside down, for 1 day. After that, place any leftover slices in an airtight container in the refrigerator for up to 3 days.