In the bowl of an electric mixer fitted with the paddle attachment, beat together almond paste and sugar on medium-high until almond paste breaks down and mixture is well combined, 3 to 5 minutes. Scrape down sides of bowl. Add almond extract and salt; beat until well combined.

With the mixer running, add egg whites, 1 teaspoon at a time, beating well and scraping down the sides of the bowl after each addition. The batter should be smooth and thick enough to pipe.

Preheat oven to 325 degrees.

Transfer batter to a pastry bag fitted with a No. 4 (1/2-inch) pastry tip. Holding pastry tip about 3/4 inch above the parchment paper on prepared baking sheet, pipe a quarter-size round of batter onto parchment paper. Lift the bag off the batter a slight sideways jerking motion of your hand, producing a flat, rather than pointy, top on the cookie. Continue piping cookies about 1 inch apart, trying to finish with as small a peak as possible to leave tops rounded and smooth. Wet a finger with warm water to smooth out any remaining peaked tops; sprinkle generously with sugar. Let stand at room temperature 15 to 20 minutes so that a thin, crisp crust develops on outside of each cookie.

Transfer baking sheets to oven and bake, rotating baking sheets top to bottom and back to front two-thirds through, until set but not browned and still gooey in the centers, 18 to 20 minutes. Transfer baking sheet to a wire rack; let cookies cool on baking sheets for 10 minutes. Transfer cookies to wire rack to cool completely.