Cut the tomatoes in half and place on a foil covered baking sheet, cut side down. Move an oven rack to the highest position and turn on the boiler. Broil 5 to 15 minutes, until the tomato skins are blackened. Move the pan around every few minutes to ensure equal roasting.
Remove from oven and cover with foil, letting the tomatoes cool and steam for 10 minutes. Remove the skins from the tomatoes and transfer to a high speed blender.
Melt the butter in a frying pan and sauté the onions, garlic, oregano, salt, pepper, and crushed red pepper for about 5 minutes or until the onions are translucent.
Add to a blender along with basil leaves, yogurt, broth, and shredded cheddar cheese. Blend on high until smooth (if using all fresh tomatoes, you might have to do this in batches).
Pour into a large pot and simmer for 10 to 15 minutes on medium-low heat. Add more salt to taste.