Make salad dressing. Whisk together lime juice, fish sauce, Thai chili sauce, brown sugar, and olive oil until sugar is dissolved. Combine Thai chili, shallots, and lemongrass in a bowl and pour dressing on top. Let sit for 10 minutes.
Heat 1 tablespoon oil in a large saute pan over high heat. Add shrimp and cook until pink and firm. Season shrimp with salt and pepper and pour half of the dressing, tossing to coat. Simmer for 1 more minute, then remove from heat.
Assemble salads. Place romaine lettuce on plates and top with sliced cucumber, cherry tomatoes, carrots, mint, and cilantro. Place shrimp on top and garnish with toasted coconut and crushed peanuts. Serve with extra dressing on the side.
Notes
The salad dressing can be prepared up to 3 days in advance.