Wash the zucchini well and dry them with paper towels. Trim off the ends and quarter the zucchini lengthwise.  Slice off the top 1/4 to 1/2 inch of the soft seed core by running a sharp knife down the length of each quarter; it's alright if some of the seeds remain.  Arrange the zucchini, cut side up on a baking sheet lined with paper towels.  Sprinkle with about 1/2 teaspoon salt and set aside for 10 minutes.
Blot the quarters dry with the paper towels.

Heat a gas grill to medium-high heat.

In a small bowl, whisk 2 tablespoons of the oil, the vinegar, thyme, lemon zest, 1/4 teaspoon salt and 1/8 teaspoon pepper.  Toss the zucchini with the remaining 1 teaspoon of olive oil.
Set the zucchini cut side down on the grill and cook, flipping occasionally, until it browns and softens but doesn't turn mushy; 6 to 8 minutes.

Cut the zucchini into 3 inch pieces and transfer to a medium bowl. Whisk the vinaigrette again and drizzle over the zucchini.  Sprinkle on the Parmigiano-Reggiano, toss well, taste and adjust seasonings accordingly.
Serve immediately.