Heat a large skillet with a tight-fitting lid over medium to medium-high. Add about 4 tbsp. oil, four turns of the pan. Add the zucchini, eggplant, onion, thyme, chile and garlic; season with salt and pepper. Cook, stirring often, until the vegetables are light golden brown, 7 to 8 minutes. Add the orzo and toss until coated, 1 to 2 minutes. Add the stock and bring to a boil. Cover, reduce the heat to medium-low and cook until the liquid is absorbed, 10 to 12 minutes. Stir in the cheese and parsley.
Heat a large cast-iron skillet over medium-high to high. Add 3 tbsp. oil, three turns of the pan. When the oil ripples, add the shrimp and oregano. Season and cook, tossing often, until the shrimp is firm and opaque in the center, about 5 minutes. Add the vermouth, lemon zest and juice.
Divide the orzo mixture among shallow bowls; top with the shrimp.