Place egg whites in a small bowl; let stand at room temperature for 30 minutes. Line 2 cookie sheets with parchment paper, plain brown paper or foil; set aside.
In a free-standing electric mixer or food processor, break up and crumble almond paste. Add 1 cup sugar. Beat or process until combined. Add egg whites, about half at a time, beating or processing until mixture is stiff (but will not form peaks) and sugar is dissolved.
Spoon egg white mixture into a pastry bag fitted with a 1/2-inch round tip. Pipe into 1-inch rounds (about the size of a quarter coin) 2 inches apart onto the prepared cookie sheets. (Or, place the egg white mixture in a resealable plastic bag; seal bag and snip off a corner of bag. Pipe as above.) Sprinkle each cookie with additional sugar.
Bake in a 375 degree F oven for 10 to 12 minutes or until set and lightly browned. Remove from oven. Cool completely on cookie sheet. Carefully peel cookies from paper or foil. (To store, layer cookies between waxed paper in an airtight container; cover. Store at room temperature up to 5 days). Makes about 52 cookies.