Special equipment: a tube pan
Make the cake: Preheat the oven to 350 degrees F.
Combine the flour, 3/4 cup of the sugar, and the salt in a medium bowl and whisk to combine.
In a large bowl, combine the egg whites with the cream of tartar and the vanilla. Beat with an electric mixer on medium speed until frothy. Increase the speed to high and gradually add the remaining 3/4 cup sugar, a few tablespoons at a time. Beat until the egg whites form glossy, medium-stiff peaks.
Sift about one-third of the flour mixture onto the egg whites and fold in with a large rubber spatula until almost completely incorporated. Continue sifting and folding in the flour mixture until all of the flour is incorporated.
Spoon the batter into an ungreased tube pan, gently smoothing the top. Bake until golden brown and the top springs back when gently pressed, 35 to 40 minutes. Turn the pan upside down on a cooling rack and cool completely.
Run a thin knife around the edge of the cake, then invert to release from the pan.
Make the compote: Combine the sugar, cinnamon stick, grapefruit peel and 3/4 cup water in a small saucepan. Bring to a boil over medium-high heat, then reduce the heat to a brisk simmer and cook until syrupy, 10 to 15 minutes. Stir in the vanilla.
Meanwhile, use a serrated knife to cut the peel and pith off of the grapefruit. Working over a bowl, use a sharp paring knife to cut between the membranes and release the grapefruit segments. Pour the sugar syrup over the grapefruit segments and let stand at room temperature for at least 5 minutes and up to 1 hour (the longer the compote sits, the more it infuses). Serve immediately or refrigerate for up to 3 hours.
Serve the angel food cake topped with grapefruit compote.