In an 8- to 10-inch frying pan, combine onion, white vinegar, 1 tablespoon of the butter, ground chili, and cumin seed. Stir often over high heat just until sizzling, 2 to 3 minutes.
Add broth and boil until reduced to about 1/2 cup, 3 to 4 minutes.
Scrape mixture into a blender. Whirl until very smooth, then whirl in remaining 6 tablespoons butter, in chunks.
Place cornbread, toasted side up, on plates and garnish equally with avocado slices. Put eggs on cornbread and spoon chili sauce over them.