Prepare the filling: Heat the olive oil in a large non-stick frying pan over medium heat. Add the diced onion and green bell pepper, and sauté for 3–4 minutes, stirring occasionally, until the onion turns translucent and the pepper softens slightly — you'll smell a sweet, earthy aroma. Add the diced chicken or turkey ham and cook for another 2 minutes, until it begins to brown at the edges and the pan sounds drier. Transfer the mixture to a bowl and set aside. Wipe the pan clean with a paper towel.

Whisk the eggs: In a mixing bowl, crack the eggs and 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 should see tiny bubbles across the surface. This aerates the eggs, giving you a lighter texture.

Melt the butter: Return the clean pan to medium-low heat. Add the unsalted butter and let it melt slowly, tilting the pan to coat the bottom evenly. The butter should foam gently but not brown — if it sizzles loudly, your heat is too high. When the foam subsides and the butter smells nutty, you're ready.

Pour and set the eggs: Pour the whisked eggs into the centre of the pan. Let them settle for about 10 seconds, then use a silicone spatula to gently push the cooked edges toward the centre, tilting the pan to let the uncooked egg flow to the edges. Repeat this 3–4 times over the next 2 minutes. The eggs should be mostly set but still look slightly wet and glossy on top — like a custard that's almost done.

Add the filling and cheese: Spoon the cooked filling evenly over one half of the omelette. Sprinkle the grated cheese over the filling. Let the omelette cook undisturbed for another 30–40 seconds, until the cheese begins to melt and the eggs look fully set but still soft — the surface should be matte with no visible liquid egg.

Fold and slide: Using your spatula, carefully fold the empty half of the omelette over the filling. Tilt the pan and slide the omelette onto a warmed plate, seam side down. The exterior should be a pale golden colour with no browning, and a slight wobble when you nudge the plate. Garnish with snipped chives and serve immediately.