Heat a ridged grill pan over medium-high heat, or a grill to medium-high (400° to 450°). Halve lemons crosswise, oil cut sides with a few drops of olive oil, and sprinkle with a pinch of salt. Grill lemons, cut side down, until well browned, 1 to 3 minutes. Let cool slightly, then squeeze into a small bowl. Add salt, sugar, and pepper; whisk to blend. Whisk in 6 tbsp. olive oil. Taste and adjust seasonings.
Quarter papayas and scoop out seeds and fibrous bits with a soup spoon. Peel quarters with a sharp paring knife, then halve them crosswise. Slice into 1/2-in.-thick pieces (about the same size as orange sections). Put in a large bowl.
Stack basil leaves and thinly slice. Add onion, radishes, basil, orange segments, and 1 1/4 cup macadamias to bowl. Toss gently with dressing.
Sprinkle with remaining macadamias and the flowers. Serve within 30 minutes.