Heat the oven to 350 °F and grease a 9x13-inch baking dish. The dish should be fully coated so the creamy edges release easily after baking.

Melt the butter with the diced onion over medium-low heat and cook until the onion turns soft and translucent, then stir in the thyme and flour. The flour should lose its raw look after 1 to 2 minutes and the mixture should look pale blond, not browned.

Lower the heat and whisk in the milk gradually, adding a little at a time until the mixture is smooth before each new addition, then stir in the chicken broth, salt, and black pepper. The sauce should thicken as it comes to a boil and coat the back of a spoon after 1 minute of bubbling.

Layer one-third of the sliced potatoes in the baking dish, scatter over one-third of the ham, and spoon over one-third of the sauce. Repeat the layers twice more, finishing with sauce on top so the upper potatoes stay moist while baking.

Cover the dish and bake for 50 minutes, then uncover and continue baking for 25 to 35 minutes more, adding the cheddar during the last 15 minutes if using. It is ready when the edges bubble in slow pops and a knife slides through the center potatoes with very little resistance.

Let the casserole rest for at least 15 minutes before serving and add parsley if you like. The resting time helps the sauce settle so the layers hold together instead of running across the plate.