Melt the sugar in a saucepan over medium heat until amber. Whisk in the butter, then slowly add the heavy cream. The mixture will bubble, so pour carefully. Stir in salt and let the caramel cool until thick but pourable.
Mix the cookie crumbs, melted butter, brown sugar, and salt until the texture looks like wet sand. Press firmly into a 9-inch springform pan, then bake at 325°F for 8–10 minutes. Let the crust cool completely.
Beat the softened cream cheese until smooth. Add brown sugar, sour cream or Greek yogurt, vanilla, salt, and cooled caramel. Mix until combined. Add the eggs one at a time on low speed, mixing just until smooth.
Pour the cheesecake filling over the cooled crust. Tap the pan gently to release air bubbles. If desired, drizzle a little extra caramel over the batter and swirl it lightly with a knife.
Bake at 325°F for 55–75 minutes, until the edges are set and the center still jiggles slightly. For a smoother cheesecake, place a pan of hot water on the lower oven rack while baking.
Turn off the oven, crack the door open, and let the cheesecake rest inside for 30–60 minutes. Then, move it to the counter and let it cool to room temperature.
Cover the cheesecake and refrigerate for at least 6 hours, or overnight for the best texture. The cheesecake should be fully chilled before slicing.
Before serving, spoon salted caramel over the top and sprinkle lightly with flaky sea salt. Slice with a hot, clean knife for neat pieces.
