Preheat the oven to 350°F (175°C). Line a 9x9-inch baking pan with nonstick aluminum foil or parchment, leaving an overhang for easy removal; spray with nonstick spray if using regular foil.
Make the crust: in a mixing bowl, beat 1 cup room-temperature butter and 3/4 cup powdered sugar on medium speed until light and creamy, about 1–2 minutes.
Reduce mixer speed to low and add 1 teaspoon vanilla, 1/2 teaspoon salt, and 2 cups all-purpose flour. Mix just until the flour is incorporated and a dough forms.
Press the dough evenly into the prepared pan and bake for 20 minutes, or until the edges are lightly golden. Remove from oven and cool completely in the pan.
Make the peanut butter filling: in a bowl, beat 1 cup smooth peanut butter with 1 cup powdered sugar and 1 teaspoon vanilla until smooth and spreadable.
Spread the peanut butter filling evenly over the cooled crust and smooth the top with an offset spatula. Chill in the refrigerator for about 30 minutes to firm up.
Prepare the chocolate topping: gently melt chocolate chips with a small splash of heavy cream in a microwave-safe bowl, heating in 15–20 second intervals and stirring between each until smooth; alternatively use a double boiler.
Pour or spread the melted chocolate over the chilled peanut butter layer and refrigerate briefly until the chocolate is set.
Once set, lift the bars from the pan using the foil overhang, cut into squares, and serve.