To make Greek vegetables, begin by adding olive oil to a hot pan. Add onion bulbs, and let them become translucent. Add artichokes, red pepper, fennel, mushrooms, cherry tomatoes, and white wine and let simmer for 15 minutes. Once the liquid is reduced, your vegetables are ready to be served.
In another pot, make the pepper coulis. Heat a pan and add the oil. Add the garlic and the peppers to the pan. When the peppers become soft, add the thyme and chicken stock. Bring to boil, reduce heat, and let the peppers become very soft. Remove the thyme sprigs, put the mixture in a blender, and puree the peppers into a sauce.
To prepare lobster, remove claws and cut lobster in half, lengthwise. Remove the eyes and any discolored pieces. Season with salt and pepper. Add a small amount of olive oil to the lobster flesh to prevent sticking to grill. Place each half flesh down, and grill for 4 to 5 minutes.
While lobster is cooking, prepare the butter that will be served with the lobster. The butter should be room temperature. Add tarragon, garlic, shallots, lemon juice, salt, and pepper. Cream all ingredients together.
To prepare the claws, boil a pot of water on stove and steam claws for 5 minutes. Quickly remove claws and cool claws in ice. Once the claws are cooled, remove the meat from the shells.
To Assemble: Place Greek vegetables in a serving dish with lobster claws on top, and red pepper coulis on side as a dipping sauce.
For the grilled lobster, place lobster on serving dish, and add butter mixture on top. Dish is ready to be served.