Brown the butter in a saucepan until fragrant and nutty, then let it cool 5–10 minutes.
Beat cooled brown butter with both sugars until smooth; add egg and vanilla and mix.
Stir in black sesame paste until evenly combined.
Whisk flour, baking soda, and salt; fold into wet mixture until just combined.
Chill dough 30 minutes if soft, then scoop tablespoon-sized balls.
Roll balls in toasted black sesame seeds and place 2 inches apart on lined baking sheets.
Bake at 350°F (175°C) for 9–11 minutes until edges set; cool on wire racks.