Handle the cakes gently when shaping them, and pan-fry until the crust is brown and crisp.
In a small bowl, combine the mayonnaise, gherkins, lemon juice, hot sauce, 1 Tbs. garlic, oil, and 1/4 tsp. pepper. Refrigerate until ready to use.
In a large bowl, combine the shrimp, panko, onion, egg, the remaining 1 Tbs. garlic, basil, 1/4 tsp. salt, and 1/4 tsp. pepper, and let sit for 10 minutes. Using wet hands and a 1/4-cup measuring cup, form about 12 cakes and refrigerate for at least 30 minutes.
Position a rack in the center of the oven. Place a rimmed baking sheet on the rack and heat the oven to 200 degrees F.
In a large nonstick skillet, heat 1/4 to 1/2 cup olive oil over medium heat, until shimmering. Fry half of the cakes until deep golden brown, about 3 to 4 minutes per side. Transfer to a paper-towel-lined plate to drain, then onto the baking sheet in the oven to keep warm. If necessary, skim the skillet. Then, add the remaining 1/4 to 1/2 cup olive oil, heat until shimmering, and fry the remaining cakes. Serve with the aïoli.