Preheat the oven to 450°F. In a large bowl, toss the thinly sliced potatoes with olive oil, salt, and pepper until evenly coated.
In a medium saucepan, combine the heavy cream, garlic, and Dijon mustard. Whisk until smooth, then heat over medium until the mixture is warm and just beginning to bubble. Remove from heat.
Line a 9x13 sheet pan with parchment paper. Arrange half of the sliced potatoes in an even layer on the pan. Sprinkle half of the diced ham and half of the shredded cheddar cheese evenly over the potatoes.
Layer the remaining potato slices on top, then carefully pour the warm cream mixture over the entire dish, making sure it seeps into the layers.
Bake for 20 minutes covered. Then, remove the foil and sprinkle the remaining cheddar cheese and the Parmesan cheese on top. Return to the oven and bake uncovered for an additional 30 minutes, or until the potatoes are fork-tender and the top is golden and crispy.
Let the dish rest for 10 minutes before serving. Finish with a sprinkle of sliced scallions or chopped chives for a fresh, flavorful finish.