Trim off the ends and quarter the zucchini lengthwise. Slice off the top half-inch of the seed core with a sharp knife; it’s okay if some seeds remain. Place the zucchini, cut side up, on a baking sheet lined with paper towels. Sprinkle with kosher salt (about half a teaspoon per pound of zucchini) and set aside for 10 minutes. Blot the quarters dry with paper towels.
Heat a gas grill to medium high or prepare a medium-hot charcoal fire. In a small bowl, whisk 2 tablespoons of the oil, the vinegar, thyme, lemon zest, ¼ teaspoon salt, and ¼ teaspoon pepper.
Toss the zucchini with the remaining 1 teaspoon olive oil. Set the zucchini cut side down on the grill and cook (if using a gas grill, keep the lid closed), flipping occasionally, until it browns and softens but doesn't turn mushy, 6 to 8 minutes. Cut the zucchini into 3-inch pieces and put in a medium bowl. Whisk the vinaigrette again and drizzle over the zucchini.
Sprinkle with the parmesan, toss well, adjust the seasonings to taste, and serve right away.