Place potatoes into a large pot and cover with salted water; bring to a boil. Reduce heat to medium-low and simmer until tender, 10 to 15 minutes.

Meanwhile, preheat the oven to 350°F (175°C). Grease a 1 ½-quart baking dish.

When the potatoes are almost finished, melt 3 tablespoons butter in a skillet over medium heat. Add ham and onion; cook and stir until the onion has softened and turned translucent, about 5 minutes. Remove from the heat.

Drain potatoes; add to the ham mixture and stir to combine. Transfer to the prepared baking dish.

Melt remaining 4 tablespoons butter in a saucepan over medium heat. Whisk in flour and stir until the mixture becomes paste-like and light golden brown, about 5 minutes.

Gradually whisk milk into the flour mixture; season with salt and black pepper. Continue cooking and stirring until thickened, about 2 minutes. Reduce heat to medium-low and stir in Cheddar cheese until melted.

Pour cheese sauce over ham and potatoes. Sprinkle bread crumbs over top.

Bake in the preheated oven until sauce is bubbly and browned, 25 to 30 minutes.