Bring a large pot of water to a boil. Prepare an ice-water bath with lightly salted water. Add salt to boiling water, return to a boil, and add beans or peas. Cook until just tender but still crunchy, 1 to 2 minutes. Drain and immediately transfer to prepared ice-water bath to cool; drain.
Place tomatoes and beans or peas in a large bowl; season with salt and pepper. Add basil and toss to combine.
In a shallow bowl, beat together milk and eggs. In another shallow bowl, mix together flour, cornstarch, salt, pepper, and cayenne pepper. Dip each crab in milk mixture to coat, and then dredge completely in flour mixture.
Set a wire rack over a parchment paper-lined baking sheet. Heat peanut oil in a large skillet over medium-high heat until it reaches 350 degrees on a deep-fry thermometer. Season crabs with salt and pepper. Working in batches, fry crabs until deep golden brown, 3 to 5 minutes. Transfer to prepared wire rack. Continue frying until all crabs have been cooked.
Drizzle salad with olive oil; toss to combine. Divide salad evenly between 4 plates and top each with 2 fried crabs. Serve immediately with garlic mayonnaise.