Line a 9x9-inch pan with parchment paper, leaving overhang on the sides for easy lifting. Lightly grease the parchment with butter or nonstick cooking spray.
Melt the butter in a large pot set over medium-low heat. Continue cooking, swirling often, until it foams, then turns golden and smells nutty with browned bits on the bottom, 4 to 6 minutes. Watch closely so it doesn't burn.
Reduce the heat to low. Add 7 cups of mini marshmallows and stir until fully melted and smooth, 1 to 2 minutes. Remove from heat.
Stir in the vanilla, peanut butter, and salt until just smooth. Add the rice cereal and fold gently until evenly coated. Add the remaining 2 cups mini marshmallows and fold once more until combined.
Immediately transfer to the prepared pan. Press down gently and evenly with a buttered spatula or your hands (don't compact too hard, or they'll be dense). Finish with flaky salt on top.
Cool for at least 30 minutes, then lift out and cut into 12 squares.