1. Make the mayonnaise: In the bowl of a food processor fitted with the blade attachment, combine the yolk and mustard. Process until the mixture is combined well. Add a few drops of lemon juice and then, with the processor still running, slowly add the oil until completely combined. Add the rest of the lemon juice and season with salt and pepper. Set aside.
2. Avocado: Halve each avocado and remove the pits. Using a large spoon, scoop the flesh out gently and slice each half into ¼”–thick slices.
3. Preheat broiler. Toast buns on both sides until golden brown.
4. Buns: Spread approximately 2 tablespoons mayonnaise on each bun. (Leftover mayonnaise will stay fresh in the refrigerator for 4 days.) Cover one side of each bun with avocado slices, one half per bun.
5. Assemble: Top the avocado with a layer of (drained) pickled radishes. Clutch each bun slightly and place side by side. Distribute the crabmeat equally among the buns. Sprinkle with lemon zest and juice, salt and pepper, and garnish with fennel fronds. Serve immediately.

Pickled Radish: Slice the radishes into ¼”– thick disks. In a non reactive saucepan, bring the remaining ingredients to a boil. Put the radishes in a clean glass jar and pour the hot liquid over them. Cover and let cool to room temperature, then refrigerate.