Preheat the grill; place the rack about 12cm from the heat.
Whisk eggs in a bowl. Add cream, Tabasco, basil, oregano, black pepper and 1 tablespoon of the grated Parmesan cheese.
Heat oil in a frypan over medium-high heat. Melt the butter in the frypan and swirl it around to coat the pan evenly. Place the oysters in the frypan and brown on both sides, about 1 minute on each side. Let the liquid reduce a bit, about 30 seconds longer.
Slowly pour the egg mixture over the oysters while keeping the oysters evenly distributed in the pan. After about 30 seconds shake the pan slightly but do not stir. After about 3 minutes when the bottom and sides of the eggs begin to set sprinkle the remaining cheese on top and place the pan under the grill.
Broil until the eggs begin to puff around the edges and the top is nicely browned, 5 to 7 minutes. Remove from the oven; sprinkle with chopped parsley. Serve immediately from the frypan in wedge-shaped slices.