Cut peel and white membrane from grapefruit. Working over a strainer set over a bowl, cut between inner membranes and fruit to release segments into strainer. Squeeze juice from membranes into bowl (discard membranes); you need about 1/4 cup juice (reserve any remainder for other uses).
Add lemon juice, mint, shallot, and honey to grapefruit juice, and mix.
Peel and pit avocados; cut lengthwise into 1/2-inch-thick slices. Set a frisée leaf on each of 4 salad or dinner plates and arrange equal portions of grapefruit segments and avocado slices on top. Sprinkle with chopped radishes.
Spoon dressing evenly over salads. Add salt and pepper to taste.