Preheat oven to 350 degrees, with racks in upper and lower thirds. In a large bowl, stir together 1 cup creamy peanut butter, 3/4 cup sugar, 1 large egg (lightly beaten), 1/2 teaspoon baking soda, and 1/4 teaspoon salt until well combined. Stir in 3/4 cup semisweet chocolate chips and 1/2 cup roasted salted peanuts.
With moistened hands, roll dough, about 1 heaping tablespoon at a time, into balls. Place 2 inches apart on two baking sheets.
Bake until cookies are golden and puffed, 12 to 14 minutes, rotating sheets from top to bottom and front to back halfway through. Cool 5 minutes on sheets; transfer cookies to racks to cool completely.