Preheat oven to 225 degrees F (110 degrees C). Mix olive oil, white sugar, rosemary, thyme, salt, and pepper in a large resealable plastic bag. Place tomatoes in the bag, seal, and shake to coat. Arrange coated tomato halves cut side up on a baking sheet. Bake tomatoes 2 1/2 hours in the preheated oven. Remove from heat, and let cool.
In a blender or food processor, finely chop the shallots. Add vinegar and brown sugar, and process until smooth. Slowly add 1 3/4 cups oil, processing frequently, so as to thicken the mixture.
Preheat grill for high heat. Brush romaine hearts with 1 tablespoon olive oil, and season with salt and pepper.
Place romaine hearts on the preheated grill. Cook 5 to 10 minutes, turning frequently, until slightly charred but not heated all the way through. Serve warm on salad plates surrounded by tomato pieces and drizzled with the shallot dressing.