Position racks in the upper and lower thirds of the oven; preheat to 350 degrees F. Line 2 baking sheets with parchment paper. Combine the peanuts and granulated sugar in a food processor and process until finely ground and beginning to clump. Pour the mixture into a small nonstick skillet and cook over medium heat, stirring often, until lightly toasted, about 5 minutes. Pour into a large bowl and let cool.
Stir the peanut butter, brown sugar, egg and vanilla into the peanut mixture with a rubber spatula until combined. Scoop heaping tablespoonfuls of dough and arrange 1 1/2 inches apart on the prepared pans. Gently press the back of a fork on top of each piece of dough twice to flatten slightly and make hatch marks (dip the fork in water if it gets sticky). Sprinkle with sanding sugar.
Bake, switching the pans halfway through, until the cookies are just set around the edges but not totally firm, 15 to 18 minutes. Let cool completely on the pans.