Heat the oil in a nonstick pan on medium heat. Saute the onion, celery, carrot, and garlic until the vegetables start to soften. Add the broth, tomato paste, and thyme. Stir well and bring to a low simmer. Season with salt and pepper.
In a large pot of boiling water, cook the cabbage leaves until flexible, about 1 minute. Set aside.
Preheat the oven to 375 degrees.
In a large bowl, combine ground beef, Worcestershire sauce, milk, breadcrumbs, and season well with salt and pepper.
Evenly distribute the filling into each cabbage leaf, then gently roll up tight, tucking in the sides as you roll. Then place the cabbage rolls seam side down into a baking dish. Pour the sauce over top, cover with foil, and bake in the oven.
Cook until the cabbage rolls reach an internal temperature of 160 degrees using a thermometer.
Spoon 1 - 2 tablespoons of the sauce with the cabbage rolls. Serve with dinner rolls.