Trim and slice the fennel bulb into 1/4-inch lengthwise strips.
Cut the radicchio into wedges.
Slice the zucchini and summer squash in half crosswise and then slice lengthwise.
Cut the onion in half lengthwise and into 1/4-inch slices.
Toss the vegetables with 2 tablespoon of the oil and the sea salt.
Spread into a single layer on a baking sheet and roast in a preheated 450-degree oven for 12 minutes. Stir once. Remove and cool.
Sprinkle with the parsley.
Whisk together the remaining oil with the vinegar, pepper, garlic and sugar. Toss the vegetables with the dressing.
Shred the Romaine and put on a platter. Top with the vegetable mixture.
Cut the cheese into shards with a Parmesan knife. Top with shards of the cheese and garnish with parsley sprigs.
Serves 6 to 8.