About 30 minutes before making macaroons, place an egg separator over a small bowl. Crack each egg over the egg separator to separate the yolks from the whites. (Save egg yolks for another recipe.)
In a clean large bowl, place the egg whites, and let stand at room temperature up to 30 minutes.*
Heat the oven to 300°F. Line cookie sheets with a sheet of foil or cooking parchment paper.
Add the cream of tartar and salt to the egg whites. Beat with an electric mixer on high speed until foamy. Beat in the sugar, 1 tablespoon at a time; continue beating until stiff and glossy--do not underbeat. Pour into a medium bowl.
Sprinkle the almond extract and coconut over egg white mixture. With a rubber spatula, cut down vertically through the batter, then slide the spatula across the bottom of the bowl and up the side, turning the batter over. Rotate the bowl a quarter turn, and repeat this down-across-up motion. Continue folding just until ingredients are blended.
For each cookie, scoop a teaspoonful of dough, using a tableware spoon, and push it onto a lined cookie sheet with another spoon or rubber spatula, placing cookies 1 inch apart. Place 1 cherry piece on each cookie.
Bake 20 to 25 minutes or just until edges are light brown. Cool on cookie sheets 10 minutes, then remove cookies from foil to a cooling rack, using a turner. Cool cookie sheets 10 minutes between batches.