Crack the 6 large eggs into a medium mixing bowl. Add 1 cup whole milk, 1 tsp salt, 1 tsp black pepper and 1 tsp Dijon mustard (if using). Whisk vigorously until the mixture is uniform and slightly frothy, about 20–30 seconds.

Set the egg mixture aside while you prepare the rest of the ingredients so flavors meld for 5 minutes.

Prepare Ham & Cheese

Dice cooked ham into roughly 1/2-inch pieces until you have about 1.5 cups. If the ham is thick or from a large piece, shorten pieces so they fold easily into eggs.

Grate 1 cup cheddar cheese and slice 2 tablespoons of chives or green onions thinly. Reserve both for adding near the end of cooking.

Cook Mix

Heat a large nonstick skillet over medium heat. Add 1 tablespoon unsalted butter and 1 teaspoon olive oil. Allow butter to melt and foam but not brown, about 30 seconds.

Add the diced ham to the skillet and cook for 2–3 minutes, stirring, until ham is warmed and edges show light color. This renders some flavor and warms the ham so it doesn't cool the eggs.

Reduce heat to medium-low. Pour the whisked egg mixture evenly over the ham in the skillet. Allow eggs to set at the edges undisturbed for about 20–30 seconds.

Using a heatproof spatula, gently stir and fold the eggs from the edges toward the center, creating soft curds. Continue cooking, stirring occasionally, until eggs are mostly set but still moist — about 2–4 minutes depending on desired doneness.

When eggs are almost finished (still slightly wet), sprinkle the grated cheddar evenly over the surface and fold it in gently so it melts and distributes. Scatter the sliced chives or green onions over the top and fold once more.

Finish & Serve

Taste and adjust seasoning with additional salt and freshly ground black pepper to taste.

Transfer the ham & egg mix to plates or keep in the skillet on very low heat for a minute to finish melting cheese. Serve immediately with toast, English muffins, or wrapped in a tortilla for a sandwich.