Fill a large pot three-quarters full with water, then bring it to a boil over high heat.
Cut the daikon radish crosswise into 2 or 3 pieces. Using a mandolin or a very sharp knife, slice the daikon into thin ⅛-inch slices.
Add 1 tablespoon of salt to the boiling water. Add the sliced daikon and cook until they’re tender but still holding their shape, about 10 minutes. Drain the daikon into a large colander set in the sink, then gently rinse them with cold water. Once drained, pat the slices dry with clean kitchen or paper towels.
Preheat the oven to 350°F (175°C). Position a rack to the upper third of the oven. Use one-third of the butter to grease an 8 x 10-inch casserole dish. Coat one side of foil big enough to cover the baking dish with a bit of butter.
In a small saucepan over medium-high heat, heat the cream, salt, pepper, thyme, onion powder, and garlic powder until simmering around the edges. Reduce the heat to low and simmer until the liquid has reduced by about one-third, 6 to 8 minutes. Turn off the heat, stir in the remaining butter, and set aside.
In a small bowl, combine the shredded cheeses. In the prepared casserole dish, layer a third of the daikon slices, slightly overlapping. Sprinkle one-third of the cheese on top. Repeat until you have three layers of daikon slices. Sprinkle the remaining cheese on top. Pour the reduced cream mixture over the layers. Cover the gratin with prepared aluminum foil and bake until the cheese is light golden brown and the liquid is bubbling around the edges, about 45 minutes. Remove, reserve the foil, sprinkle the gratin with Parmesan cheese, and bake, uncovered, for 15 minutes more. Switch the oven setting to broil, then briefly broil until the gratin crust is deep golden brown and bubbling, 1 to 2 minutes.
Remove the gratin from the oven and cover with the reserved foil. Let stand and set for 10 minutes. Serve.