Cook whole zucchini in boiling salted water, covered, for about 10 minutes. Drain and cut in half lengthwise. Scoop out centers, leaving a 1/4-inch thick shell. Chop the center pulp.
In a skillet melt butter; sauté the sliced mushrooms. Stir in flour, salt, and oregano. Remove from heat. Stir in Monterey Jack cheese, sour cream, and the chopped zucchini. Fill zucchini shells, using about 1/4 to 1/3 cup of stuffing mixture for each. Top with grated Parmesan cheese. Place stuffed zucchini in broiler pan and broil for about 4 to 6 minutes, until hot and bubbly.
Serves 3 as a main dish or 6 as a side dish.
Calories per serving (based on 6 side dish servings): 160