Start by making the dressing. Place all the ingredients in the bowl of a food processor. Process until very smooth, scraping down the sides of the bowl a few times if necessary.
Transfer the dressing to a mixing bowl. Add the diced chicken breast and shallot, and toss to combine. Taste and season with a little extra salt and cayenne pepper if necessary.
Time to assemble your lettuce wraps! Pile chicken salad onto lettuce leaves. Top with sprouts and garnish with jalapeño and radish slices. Serve with lime wedges and go to town.