Combine the egg whites, sugar, honey, and vanilla in a bowl (or in a double boiler) and place over a large saucepan of simmering water. Cook and stir constantly for 8 to 10 minutes, until the sugar dissolves. The mixture will look milky in color.
Combine the coconut and flour in a separate bowl and toss to mix.
Remove the egg white mixture from the heat and stir in the coconut mixture. Cover and refrigerate until firm, about 1 hour or overnight.
Preheat the oven to 300 degrees.
Lightly grease 2 large baking sheets and set aside.
Remove the batter from the refrigerator and scoop with a ¼-cup (2 ounce) ice cream scoop or by the heaping tablespoon and drop onto the prepared baking sheets about 3 inches apart.
Bake 15 to 20 minutes, until golden brown but still soft and creamy in the center. Cool 5 to 10 minutes before removing the macaroons to a baking rack to cool completely.
Variation: Dip cooled cookies halfway into 6 to 8 ounces melted semisweet chocolate; let stand 15 to 20 minutes for the chocolate to set.