In a shallow skillet over medium heat, melt 1 tablespoon of the butter with 1 tablespoon of the oil.
Once hot, fry the slices of bread on both sides, adding more oil and butter as needed. Set aside, but do not cover.
If you are making the Gouda Cheese Sauce, in a medium saucepan, melt the butter over medium heat. Once hot, add the flour and stir with a whisk, breaking up all lumps. Continuously stir for 2 minutes; this will take the raw flour taste out of your sauce. Turn the heat down if it is beginning to brown. You want to keep a light-colored sauce.
Add the milk and stir occasionally until you can see the sauce beginning to thicken, then stir continually until thickened further, but still a bit pourable.
Remove the saucepan from the heat and add the cheese in two additions, stirring until the cheese has melted after each addition.
Taste and adjust any salt and pepper according to your tastes. Turn the heat to the lowest setting on the smallest burner until ready to use. Add more milk as necessary if the sauce has thickened to where it is not pourable.
Cook the eggs according to preference. I found cooking 2 at a time when the yolks need to be intact worked well since that is 2 per serving. Scrambled eggs are easy to cook several at once, then divide into servings.
To assemble, per person, layer 1 slice of bread, 2 slices of ham, 2 slices of cheese, or a ⅓ cup cheese sauce, 2 eggs cooked to order, and sliced chives.
Dig in with a knife and fork!