Preheat the oven to 350°F. Line two baking sheets with parchment paper and set them aside.
Fill a skillet about halfway with water and bring it to a gentle simmer. Use a heatproof bowl that will sit on the skillet without touching the water.
In the heatproof bowl, combine 4 egg whites, 3 cups sweetened coconut, ¾ cup sugar, 2 teaspoons vanilla, and ¼ teaspoon salt; stir until evenly mixed.
Set the bowl on the skillet (double boiler) over the simmering water. Cook, stirring frequently, for about 5 minutes, until the mixture thickens and becomes opaque.
Remove the bowl from the skillet and let the mixture cool about 30 minutes, or until cool enough to scoop.
Scoop about 2 tablespoons of the mixture for each macaroon (a medium cookie scoop or a 2-tablespoon measuring spoon works), and place the mounds about 2 inches apart on the prepared baking sheets.
Bake at 350°F for 5 minutes. Without removing the sheets from the oven, lower the oven temperature to 325°F and continue baking for 10–15 minutes more, until the macaroons are lightly browned with some darker edges.
Remove the baking sheets from the oven and let the macaroons cool on the sheets a few minutes before transferring to a rack to cool completely.