Combine the half-and-half and heavy cream together in a small bowl. Pour a 1/2 cup of the cream mixture into a 2.5- to 3-quart baking dish (you just want to cover the bottom).
Carefully combine the ham, onion, 2-1/2 cups of Monterey Jack, 2-1/2 cups of cheddar, thyme, salt, and pepper with the potatoes in a large mixing bowl until evenly combined. Be careful not to break the potatoes.
Evenly spread half the potato mixture in the baking dish. Top with half the remaining cream mixture (about 1/2 cup). Repeat with the remaining potato mixture and cream mixture. Top with the remaining cheese.
Cover and bake for 1 hour. Uncover and continue to bake until bubbly and brown, about 40 minutes. Let the dish rest 15 minutes before serving.