Preheat the oven to 350°. Using a food processor, finely grind the pretzels, 1/4 cup brown sugar.
Transfer to a bowl, then stir in the melted butter.
Press evenly over the bottom and up the sides of a 9-inch tart pan with a removable bottom.
Bake until light brown around the edges, about 18 minutes. Let cool on a rack.
Using an electric mixer, beat the cream cheese, peanut butter and softened butter at medium-high speed until fluffy, about 1 minute.
Beat in the remaining 1/4 cup brown sugar.
In a separate bowl, beat 1 cup heavy cream with the vanilla until soft peaks form.
Fold the whipped cream into the peanut butter mixture.
With a spatula spread this gorgeous peanut butter mixture in the tart shell.
Cover with plastic wrap, then refrigerate until firm, at least 3 hours. I popped the bottom out of the sides of the tart pan. I left it the bottom on the tart pan and it was very messy when cutting.
In a small, heavy skillet, bring the granulated sugar and 3 tablespoons water to a boil, stirring until the sugar has dissolved (wash down any sugar crystals from the sides with a wet pastry brush). Boil without stirring, but swirling the pan occasionally, until golden-brown, about 5 minutes. Let cool for 2 minutes, then whisk in the chilled butter, the remaining 1/4 cup heavy cream and 1/4 teaspoon salt. Transfer the caramel to a bowl and let cool completely. Rewarm gently before drizzling over the sliced tart; serve the remainder on the side.