Preheat the oven to 300˚ F. Line a baking sheet with parchment paper.
Wash and dry the tomatoes, and slice them in half, horizontally. Lay them, face up, on the parchment paper. drizzle the oil over them, allowing oil to drizzle onto the parchment paper.
Sprinkle the tomato halves with salt, pepper and dried thyme.
Roast for 3 - 4 hours. Drizzle the balsamic vinegar over the tomatoes, and allow to cool.
Pack into a container and lay the fresh thyme twigs across them. Pour all of the excess oli from the parchment paper, now essentially a tomato oil, onto the top and add about 1 tablespoon of additional olive oil.
Store in the refrigerator for up to one month. They can also be frozen for at least six months. They can probably be frozen for longer, but I haven't personally done it. I eat them up too fast. :-).
Calories were calculated using the VeryWell Fit nutrition calculator for 12 servings.