Preheat the oven to 350°F (175°C) and position a rack in the center.
Heat a 10- to 11-inch oven-safe deep sauté pan or skillet over medium heat. Add the 1 cup unsalted butter and melt completely.
Remove the pan from the heat. Immediately whisk in the 1 cup sugar and 1 cup dark brown sugar until the sugars are dissolved and the mixture is smooth.
Whisk the 3 large lightly beaten eggs into the warm butter-sugar mixture quickly and continuously so the eggs do not scramble. Stir in the 1 tablespoon vanilla extract.
In a separate bowl, stir together 1 cup all-purpose flour, 3/4 cup unsweetened cocoa powder, 1 tablespoon instant espresso powder, and 1 teaspoon salt until evenly combined.
Add the dry mixture to the skillet and stir with a spatula or wooden spoon until just combined and there are no visible streaks of flour or large lumps.
Stir in 1 cup chocolate chips until evenly distributed through the batter.
Drop the 3/4 cup creamy peanut butter in tablespoon-sized dollops all over the top of the batter. Use a knife to gently swirl the peanut butter into the batter, leaving visible swirls.
Smooth the top lightly if needed, then place the skillet on the center rack of the preheated oven. Bake for 25 to 30 minutes, or until the edges are set and the center is just barely set and slightly jiggly.
Remove the skillet from the oven and, if desired, sprinkle additional chocolate chips on top. Let the brownie cool in the skillet for a few minutes.
Serve warm, topped with vanilla ice cream.