Season fish with 11/2 tsp. salt and 3/4 tsp. pepper. Heat 3 tbsp. oil in a large castiron skillet over medium-high heat. Cook lingcod until golden and it flakes at the touch of a fork, about 5 minutes per side. Transfer fish to dinner plates.
Meanwhile, using a vegetable peeler, shave cucumbers into long strips on one side until you start hitting seeds, then flip it over and shave opposite side, to equal about 6 cups.
Combine cucumber, onion, lemon juice and zest, basil, and mint with remaining 3 tbsp. olive oil, 11/2 tsp. salt, and 3/4 tsp. pepper; toss to combine.
Add salad to each plate, along with a lemon wedge.