Prep everything first: peel, devein and pat the shrimp dry with paper towels (moisture prevents browning), thinly slice the scallions — separate white and green parts — grate the ginger, mince the garlic (if using), and thinly slice the cucumber lengthwise then into half-moons. Juice half a lime and reserve the other half for serving. Crack the eggs into a bowl and lightly beat with a pinch of salt and a grind of pepper. Having components ready keeps cooking fast and controlled (mise en place).

Make the quick cucumber salad: put the sliced cucumber in a bowl with 2 tbsp rice vinegar, 1 tsp of the olive oil, grated ginger (about 1/2 tsp of the grated amount), 1/2 tsp sugar (optional, to balance acid), 1/4 tsp salt and 1 tbsp lime juice. Toss and let sit at room temperature while you cook the omelettes so the vinegar slightly softens the cucumber and infuses flavor. The acid brightens the cucumber and contrasts the richness of the eggs.

Season and pre-cook the shrimp: dry shrimp then season lightly with 1/4 tsp salt, a pinch of pepper, and a tiny squeeze of lime. Heat a nonstick or well-seasoned 10-inch skillet over medium-high heat and add 1/2 tsp olive oil. When oil shimmers, add shrimp in a single layer and sear 1–1½ minutes per side until they just turn opaque and curl into a loose C (overcooking makes shrimp rubbery). Remove shrimp to a plate and let rest; carrying residual heat will finish them. Quick sear develops Maillard flavor that complements the eggs.

Saute scallion whites: lower heat to medium and add the tablespoon of butter to the same pan. When butter foams, add the white parts of the scallions and the minced garlic. Sauté 30–45 seconds until fragrant and softened but not browned — this gently sweetens them and creates a base flavor for the omelette. Stir in half the cooked shrimp (reserve the rest for garnish).

Make the egg base: give the beaten eggs a final whisk (this aerates them for a silkier omelette) and add a small pinch of salt (about 1/8 tsp). Push the scallion/shrimp mixture to one side of the pan and melt a little more butter or swirl 1/4 tsp olive oil across the pan to ensure an even nonstick surface. Pour eggs into the pan; they should sizzle softly — that gentle heat cooks the eggs slowly so they stay tender rather than rubbery.

Control the set: as the eggs begin to set around the edges, use a spatula to gently pull cooked egg from the edge toward the center, letting runny egg flow to the pan surface. This technique (a folded omelette technique) creates a tender curd and even texture. When eggs are mostly set but still slightly glossy on top (about 1–2 minutes), scatter the reserved green parts of the scallions, remaining shrimp, and a light grind of pepper over one half.

Fold and rest: fold the omelette in half over the filling and let it sit, off heat, for 30 seconds to 1 minute. Carryover heat finishes the eggs and warms the shrimp without overcooking. Slide the omelette onto a warmed plate; dividing into three portions now yields even servings.

Finish the salad and plate: give the cucumber salad one last toss and taste — adjust with a pinch more salt, a splash of lime, or a touch of sugar if it's too sharp. Spoon the salad alongside each omelette portion and finish with a lime wedge. Optional: a drizzle of sesame oil or a sprinkle of toasted sesame seeds or chili flakes for heat.

Serve immediately so the omelette stays tender and the salad stays crisp. Hot omelette + cool acidic salad is contrast that feels balanced and restaurant-polished.