Working on a cutting board with a juice well, cut off and discard ends from grapefruit, tangerine, orange, blood orange, lemon, and lime. With a small, sharp knife, cut peels and outer membranes from fruit and discard. Squeeze any juice from membranes into a 1-cup measure and reserve.
Cut fruit crosswise into 1/2-inch-thick slices, then cut slices into cubes, discarding seeds. Pour juice from well of cutting board into glass measure; reserve for nut-crusted sole.
Pit, peel, and dice avocado. In a bowl, gently mix citrus cubes, avocado, kumquats, chili, mint, and salt.