Preheat oven to 325 °F. Coat one or two baking sheets with cooking spray. Line with parchment paper or foil and coat again.
Place almonds, flour, cocoa and salt in a food processor or blender and process until the almonds are finely ground. Whisk together egg whites, sugar, oil, butter and vanilla in a mixing bowl. Add the flour-almond mixture and whisk until well blended.
If making cookie flutes, drop a scant tablespoonful of batter onto a prepared baking sheet. With a metal spatula or the back of a spoon, spread the batter into a circle 4 to 4 1/2 inches in diameter. Repeat with the remaining batter, leaving ample space between cookies—you will have room for about 3 cookies per baking sheet. Bake until the tops of the cookies are firm, 10 to 12 minutes. Immediately lift the parchment or foil from the baking sheet and place it on the counter. Carefully slip a flat metal spatula under each cookie to loosen it. Return the cookies to the baking sheet and place in the oven for about 1 minute to soften. To shape, wrap each cookie loosely around a wooden spoon handle to form a flute. (If making cookie cups, use a generous tablespoonful for each cookie. Spread the batter into circles 5 to 5 1/2 inches in diameter. Bake as directed above. To shape, drape the softened cookies over an upside-down glass—a narrow glass, about 2 inches in diameter, works best—and pleat the edges to make a cup.)
Transfer the shaped cookies to a wire rack to finish cooling. Before serving, dust the edges of the cookies with confectioners' sugar.
Make Ahead Tip: Store in an airtight container for up to 1 week.