Preheat the oven to 325°F. Line two baking sheets with parchment paper.
In the bowl of a standing mixer, whisk egg whites, sugar, vanilla, and salt on high speed until thick and opaque white (about 1 minute).
Use a spatula to fold the coconut and flour into the egg white mixture.
Use a 1 ½ tablespoon cookie scoop or a rounded spoon to drop compact balls onto prepared baking sheets.
Bake for approximately 25 minutes, rotating the pans during baking, until cookies are golden brown.
After 1 to 2 minutes, transfer cookies to a cooling rack and cool completely.
In the top of a double boiler, melt chocolate chips and shortening until smooth and fully melted.
Transfer melted chocolate to a shallow bowl if needed to make it easier to dip. Dip the bottom of each cooled macaroon into the chocolate, just enough to coat the bottom and up the sides a bit. Allow the excess chocolate to drip off the bottom. Place a dipped cookie on a sheet of parchment paper on a baking sheet.
Allow the chocolate to set (place the baking sheet in the fridge for a few minutes to help it set more quickly).