Shave the brussels sprouts horizontally into 1/8-inch slices with a sharp knife or mandoline. Add to a bowl. Add the Cheddar, flour, thyme, garlic and some salt and pepper to the bowl. Toss to combine. Add the mixture to an 8- by 8-inch baking dish, packing it in. Pour over the heavy cream.
In a small bowl, combine the panko, Parmesan and oil and pour over the brussels sprouts mixture.
Bake uncovered until the brussels sprouts are tender, the sauce is bubbling and the top is golden, 25 to 30 minutes. Garnish with the parsley and serve.