Bring a large pot of water to a boil over high heat. Sprinkle generously with salt. Trim the base of the cauliflower so the stem is flat to the head but without cutting off any florets. Blanch the cauliflower until just beginning to soften, 8 to 10 minutes. Remove to a plate to cool slightly.
In a small skillet, heat the butter until melted. Add the breadcrumbs and 1/2 teaspoon salt and toast until golden brown, about 3 minutes. Set aside to cool completely. Stir in the Parmesan and basil.
Gently begin to create space in between the florets and stuff each space with a log of Cheddar. Continue until all the pieces of cheese are used. Press the toasted breadcrumbs on top of the cauliflower. Place in a small baking dish and bake until the cauliflower is tender all the way through and the breadcrumbs are golden, about 20 minutes.