Preheat oven to 350 degrees. Line a buttered 8-inch square baking pan with parchment, allowing a 2-inch overhang. Butter lining (not overhang).
Stir together 1 tablespoon of sugar, the condensed milk, coconut, egg white, and 1/4 teaspoon of the vanilla in a bowl.
Put butter and both chocolates in a heatproof bowl set over a pan of simmering water; stir until melted. Let cool slightly.
Whisk together flour, baking powder, and salt in a bowl.
Whisk remaining 3/4 cup sugar into chocolate mixture. Add eggs; whisk until mixture is smooth. Stir in remaining 2 teaspoons vanilla. Add flour mixture; stir until well combined.
Pour 1/3 of chocolate batter into prepared pan. Spread evenly with an offset spatula. Drop dollops of coconut mixture (about 1 tablespoon each) on top of batter, spacing about 1 inch apart. Drizzle remaining batter on top, and gently spread to fill pan. Drop dollops of remaining coconut mixture on top.
Gently swirl coconut mixture into batter with a butter knife, running the knife lengthwise and crosswise through layers. Bake until a cake tester inserted into brownies (avoid center and edges) comes out with a few crumbs but is not wet, 35 to 40 minutes. Let cool slightly in pan, about 15 minutes. Lift out; let cool completely on a wire rack before cutting into squares. Brownies can be stored between layers of parchment in an airtight container at room temperature up to 2 days.