In a mixing bowl of a stand mixer fitted with the paddle attachment, combine completely softened cream cheese blocks and sugar and beat on medium-low speed, about 30 seconds.
If the cream cheese can't be whipped smoothly, turn off the mixer immediately and let it continue softening at room temperature.
Add one egg at a time to the cream cheese mixture and beat until everything is incorporated, about 1 minute. The mixture will become smoother after each added egg.
Then add heavy cream, salt, and vanilla extract and continue beating until the mixture is smooth approximately 30 seconds. Scrape down the bowl's sides with the spatula a couple of times during the mixing process.
Carefully add flour into the cream cheese mixture and mix until everything is fully incorporated approximately 30 seconds. The batter should be smooth, silky, and free of a lump at this point.
Grease the bottom and the sides of the baking pan with softened butter.
Measure and cut two big pieces of parchment paper to line the pan. I cut 15-in-square-parchment-papers for my 8" baking pan. Remember to lightly grease between the parchments to make them stay in place easily.
Because of this, classic Basque burnt cheesecake won't have smooth sides like the other cheesecake recipes.
Pour batter into the pan, and use a spatula or back of a spoon to smooth out the top if necessary.
Bake cheesecake for 40 minutes. The cheesecake should have a dark brown/caramelized effect on top.
Let the cheesecake cool down completely before serving. The top of the cheesecake will sink from 1 to 2 inches during this process. You can either serve it at room temperature or straight from the refrigerator.
Notes
See the sections above for valuable tips for making the perfect, foolproof cheesecake and storing and cooling instructions.