In a bowl, mix 1/4 cup champagne vinegar, 2 tablespoons minced shallot, and 1 teaspoon salt. Let stand 5 minutes.
Whisk in 3/4 cup extra-virgin olive oil.
Rinse 2 small green and 2 yellow zucchini (8 ounces total) and trim off ends. Using a mandoline, slice squash lengthwise 1/16 inch thick, discarding first and last slices from each.
In single-layer batches, steam squash over boiling water until barely tender when pierced, about 1 1/2 minutes. Arrange warm slices in a single layer in a cooking parchment-lined baking pan and brush tops with vinaigrette. Cover and chill until cold, at least 20 minutes.
Arrange slices, mixing colors evenly, on six large plates. Drizzle with a little more vinaigrette and sprinkle with 1/3 cup toasted pine nuts (2 oz.) and 1/2 cup slivered fresh basil leaves.
Shave Parmigiano-Reggiano cheese over servings to garnish, and drizzle with a little more olive oil.