Preheat an oven to 425°F. Preheat a grill (preferably charcoal) to medium-high heat.
Add polenta, water, 1 1/4 cups milk, garlic, 1 1/2 teaspoons kosher salt and 1/4 teaspoon black pepper to a shallow pan or dutch oven, whisking to combine. Cover and bake until polenta has absorbed most of the liquid (30 mins), stirring halfway through. Finish the polenta by stirring in butter, 1/4 cup milk, cheddar and Parmesan. Season to taste with additional salt and pepper. If the polenta firms up, stir in a little more milk to loosen up. Cover and keep warm until ready to serve.
Meanwhile, drizzle the radicchio with olive oil, so it's evenly coated. Season all over with salt. Grill the radicchio, cut-side down, beside the brats. Cook until the radicchio is charred (about 3 minutes), then flip over and cook for 1 minute more. The brats will need 8 to 9 minutes total and a flip halfway through. Thinly slice the radicchio, then toss with lemon juice and 1 tablespoon of olive oil. Season to taste with kosher salt. Also slice the brats.
To serve, top polenta with radicchio salad and sliced brats. Finish with a sprinkling of small basil leaves (or torn big leaves). Enjoy immediately!