Bring a large pot of water to a boil over medium-high heat; prepare a bowl of ice water. Season the boiling water with salt. Add the carrots and cook for 1 minute. Add the green beans to the pot with the carrots and continue cooking both for 3 minutes. Drain the vegetables in a colander, then immediately transfer to the bowl of ice water to stop the cooking. Remove from the water and pat dry.
Preheat a large (2-burner) grill pan over medium-high heat.
Put the carrots and green beans on a baking sheet with the mushrooms, zucchini, squash and radicchio. Drizzle 3 tablespoons of the olive oil on the vegetables and toss to coat. Season lightly with salt and pepper.
Grill the vegetables, in batches if necessary, until tender and well-marked, turning them over halfway (about 3 minutes for the green beans and carrots and 8 minutes for the mushrooms, radicchio, zucchini and squash). Transfer the vegetables to a cutting board and chop into bite-size pieces. Slice the radicchio crosswise into shredded pieces.
Add the smaller, whole leaves of Bibb lettuce to a large serving bowl. Top with the chopped vegetables.
Whisk together the shallot, lemon zest and juice and some salt and pepper in a small bowl. Add the remaining 3 tablespoons olive oil and whisk until emulsified. Drizzle the vinaigrette over the salad. Top with grated cheese. Serve warm or at room temperature.