Make the crust first. Preheat oven to 350 degrees. In a food processor, pulse cookies until finely ground (you should have about 1 1/2 cups). Add sugar, salt, and butter and pulse until it looks like wet sand. Scrape the sides if necessary.
Use the bottom of a glass or measuring cup to firmly press crumb mixture into the bottom and halfway up the sides of a 9 (or 9-1/2) inch springform pan.
Bake until crust is dry and set, about 10 minutes. Let cool completely. Stick it in the freezer or the fridge to speed this up if you want.
In a large bowl or stand mixer, beat cream cheese on high until fluffy, about 3-4 minutes. Add the can of Dulce de Leche; beat until smooth, scraping down sides.
In a separate medium bowl, beat cream and powdered sugar on high until soft peaks form, about 3 minutes. With a rubber spatula, stir about one-third of the whipped cream into cream cheese mixture, then fold in remainder. It should be fully incorporated, but try hard not to overwork it.
Transfer to crust and smooth the top with a spatula. Cover tightly and refrigerate until firm, 3 hours (or up to 2 days).
Peel* and slice the peaches. In a large skillet, melt 1 tablespoon of butter over medium heat. When the butter is hot, add as many peach slices (cut side down) that will fit. Work in batches if you need to--the peaches should not touch each other. Leave plenty of space or they will steam and not caramelize. Cook the peaches for about 1 minute without stirring.
Sprinkle the peaches with 1-2 tablespoons brown sugar, enough to cover the peaches in the pan. Flip each peach so that the sugar is on the bottom. Cook for 3-4 minutes without stirring. Sprinkle with more brown sugar and flip to brown the other side.
The brown sugar should bubble up around the peach. If there is too much peach liquid around the peach, then it will not caramelize or bubble. Move it to a drier spot on the pan and add more sugar.
Remove the caramelized peaches to a wax-paper lined sheet pan. Let cool complete. I put mine in the freezer for about 10-15 minutes.
Arrange the caramelized peaches on the cheesecake just before serving. If they don't all fit on top, serve the remainder on the side.