3 to 4 tablespoon extra-virgin olive oil, more to taste
1/4 teaspoon salt, more to taste
2 pounds sweet or hot Italian sausages
1 lemon, halved
Basil leaves, for serving
Light or heat up the grill. Halve radicchios lengthwise through stem end, and toss with 1 tablespoon olive oil (or more if you need it to coat) and 1/4 teaspoon salt.
Put radicchios, cut side down, and sausages on the grill and cover. Let cook for 2 to 3 minutes, until outer radicchio leaves are dark brown. Transfer to a cutting board and let cool while sausages finish cooking, about 5 minutes more.
When cool enough to handle, thinly slice radicchios. The outsides should be charred, and the insides should still be crisp and raw. Combine in a bowl with another 2 tablespoons olive oil and the juice of half a lemon. Taste and correct seasonings, adding lemon juice, salt and olive oil to taste.
Spread radicchio on a platter and top with grilled sausages. Garnish with basil and serve.