Bring butter, brown sugar, and corn syrup to a boil over moderate heat, stirring. Add flour and cook, stirring constantly, until dough is slightly thickened, about 1 minute. Stir in coconut and cool to room temperature, about 45 minutes.
Preheat oven to 375°F. Crumple foil to form a solid cone (about 6 inches long and 3 inches across widest part) for shaping cookies.
Drop 2 (1-tablespoon) portions of dough 6 inches apart on a buttered baking sheet and pat each into a 5-inch round with your fingertips. Bake in middle of oven until golden, 6 to 8 minutes (cookies will spread to about 6 inches).
Let cookies stand on baking sheet until just firm enough to hold their shape, about 2 minutes. Gently loosen cookies from baking sheet with offset spatula, then turn over so smooth sides are facing up. Roll 1 cookie onto cone and cool on cone until cookie holds cone shape, about 15 seconds, then carefully slip off onto a rack to cool completely. Shape remaining cookie in same manner. (If cookies become too brittle to roll onto cone, return baking sheet to oven for a few seconds to soften.) Make and shape more cookies with remaining dough.
Put 3 small scoops ice cream into each cone and put cones on plates.
