The French term "a la grecque" ("in the Greek style") refers to this salad's marinade of lemon juice, olive oil, and oregano, which is used to flavor the raw vegetables.
Combine shallots, olive oil, and dried oregano in a small saucepan, and place over medium-low heat. Let cook until shallots are translucent and very soft, about 20 minutes. if oil starts to sputter or brown, reduce heat.
Meanwhile, cut vegetables, and set each aside in its own, separate bowl: Cut cauliflower and broccoli into bite-sized florets. Trim stems from mushrooms, and cut them in half. Slice each celery stalk and carrot into 3-inch-long, thin sticks. Slice larger radishes in half.
Season each bowl of vegetables with about 1/2 teaspoon salt and 1/4 teaspoon black pepper. Add lemon juice to shallots and warm oil; stir to combine. Pour just less than 1/6 of the mixture over each bowl of prepared vegetables, keeping in mind that the mushrooms will absorb the dressing and require a little more than the other vegetables. Toss each bowl of vegetables; let stand for between 1 and 4 hours. Arrange vegetables on platter, and serve.
This salad is best prepared several hours in advance and served at room temperature; each vegetable marinates in warm dressing in its own bowl.