Roast the tomatoes in a 400 F degree oven for 10 minutes; remove and promptly cool.
Once the tomatoes are cool, peel and seed them. Cut the tomatoes into 1/2" pieces.
Place tomatoes, shallots, garlic, mustard, egg yolks and vinegar into a stainless steel bowl and mix thoroughly. Slowly add 2/3rds of the oil, add the salt and pepper, then add the basil. Add the last 1/3rd of the oil. Clean the basil from the whip. Check for proper salt and pepper seasoning.