Sauté the filling: Heat the vegetable oil in a large non-stick frying pan over medium-high heat until it shimmers. Add the diced green pepper and onion, stirring frequently. Cook for 2–3 minutes until the onion turns translucent and the pepper softens, releasing a sweet, savoury aroma. Add the diced chicken and cook for another minute, just until heated through. Transfer this mixture to a small bowl and set aside. Wipe the pan clean with a paper towel.

Whisk the eggs: Crack the eggs into a medium mixing bowl. Add the milk, salt, and pepper. Whisk vigorously for about 30 seconds until the mixture is a uniform pale yellow and slightly frothy on top — you want the yolks and whites fully combined so the omelette cooks evenly.

Melt the butter: Return the clean pan to medium heat. Add the butter and let it melt slowly, tilting the pan to coat the bottom evenly. The butter should foam gently but not brown — if it starts sizzling loudly, your heat is too high.

Pour and set: Pour the whisked eggs into the centre of the pan. Let them settle for about 10 seconds, then use a spatula to gently push the cooked edges toward the centre, tilting the pan so the uncooked egg flows to the edges. Repeat this motion 3–4 times until the eggs are mostly set but still slightly wet on top — about 2 minutes. The bottom should be a light golden brown.

Add the filling: Spoon the sautéed pepper, onion, and chicken mixture evenly over one half of the omelette. Sprinkle the grated cheddar cheese on top. The residual heat will begin melting the cheese immediately.

Fold and finish: Using your spatula, carefully lift the unfilled half of the omelette and fold it over the filling. Press down gently with the spatula to seal. Let it cook for another 30–45 seconds until the cheese is fully melted and the omelette is heated through. The edges should look slightly crisp, and you'll hear a soft sizzle.

Slide and serve: Tilt the pan over a warm plate and let the omelette slide out gently. The surface should be golden with no browning, and the interior steam should smell richly of cooked eggs and melted cheese. Serve immediately.