Toasted-Coconut Marshmallow Squares

Preheat oven to 350 °F with rack in middle. Toast coconut in a shallow baking pan in oven, stirring occasionally, until golden, 7 to 10 minutes. Oil 9-inch baking pan, then sprinkle bottom with 1/2 cup toasted coconut. Sprinkle gelatin over 1/2 cup water in bowl of mixer and let soften while making syrup. Heat sugar, corn syrup, salt, and remaining 1/2 cup water in a small heavy saucepan over low heat, stirring until sugar has dissolved. Bring to a boil over medium heat, without stirring, washing any sugar crystals down side of pan with a pastry brush dipped in cold water. Put thermometer into syrup and continue boiling, without stirring, until it registers 240 °F (soft-ball stage). Remove from heat and let stand until bubbles dissipate. With mixer at low speed, pour hot syrup into gelatin in a thin stream down side of bowl. Increase speed to high and beat until very thick, about 15 minutes. Add vanilla and coconut extracts and beat 1 minute more. Spoon marshmallow over toasted coconut in baking pan and press evenly with dampened fingertips to smooth top (it will be very sticky), then evenly sprinkle top with 1/2 cup toasted coconut. Let stand, uncovered, at room temperature until firm, about 2 hours. Run a sharp knife around edge of marshmallow and invert onto a cutting board. Cut into 3/4-inch-wide strips, then cut each strip into 3/4-inch squares. Put remaining toasted coconut in a small bowl and dredge marshmallows in it to coat completely.