Preheat the oven to 350 degrees . Using a food processor, finely grind the pretzels, 1/4 cup brown sugar and 1/8 teaspoon salt. 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, then fold into the peanut butter mixture. Evenly spread the peanut butter mixture in the tart shell. Cover with plastic wrap, then refrigerate until firm, at least 3 hours.
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.