Heat oven to 350°F. Line two baking sheets with parchment.
Cream butter, granulated sugar, and brown sugar until light and fluffy, 2–3 minutes.
Beat in egg yolks and vanilla until smooth.
Mix in flour and fine salt just until a soft dough forms. Do not overmix.
Whisk egg whites in a small bowl. Spread coconut in a shallow dish.
Scoop 1-tablespoon portions of dough, roll into balls, dip in egg whites, then coat in coconut.
Arrange on sheets 2 inches apart. Press a deep center well with your thumb or a 1/2-teaspoon measure.
Bake 8 minutes. Remove and gently re-press the wells. Bake 5–7 minutes more until edges are light golden.
Melt caramels with heavy cream over low heat, stirring until silky. Keep warm on the lowest heat.
Spoon warm caramel into the centers. Sprinkle with flaky sea salt. Cool on the sheets until set.