Whisk sugar, oil, and seasonings together in a large bowl; add shrimp and toss to coat.
Thread 2 shrimp onto each of 8 skewers, then grill just until opaque, 2 minutes per side.
Prepare grill with foil, then preheat to medium-high.
Whisk mayonnaise, 2 T. oil, vinegar, sugar, salt, and cayenne together for the dressing in a bowl. Prepare shrimp and thread on skewers.
Coat corn, asparagus, and bread with 3 T. oil, salt, and pepper. Grill corn, turning occasionally, until kernels start to char, 10 minutes. Add asparagus and cook, turning often, until tender, 3–4 minutes.
Grill bread on both sides until toasted, 1–2 minutes per side; grill the skewers.
Cut corn off cobs, chop asparagus, and cube bread.
Toss lettuces, cucumber, tomatoes, eggs, and basil together in a large bowl with remaining ingredients and prepared dressing. Serve 2 shrimp skewers on top of each salad.
Combine dressing ingredients, then whisk until blended. Season with salt and cayenne
Toss shrimp in the seasoning paste, then thread onto skewers through the tail and head.
The corn will take longer to grill than the shrimp and asparagus, so start cooking
After tossing the salad with the dressing, serve it right away so it stays crisp