The peppers should stand upright. If not, place them on a cutting board, and trim about 1/8-inch off the bottoms to even them out so they can stand without falling over.
Cut the top of each pepper about 1/2 inch or a little bit more.
Remove and discard the stems, then chop all the remaining flesh from the tops.
Scoop out the seeds and as much of the membranes as you can.
Now, place the peppers in a large microwave-safe bowl with a cup of water.
Cover with plastic wrap and heat the microwave on high power for about 10 minutes.
Carefully uncover and let them sit until ready to stuff.
Heat 2 tablespoons of olive oil in a large frying pan over medium-high heat.
Spread the beef in an even layer and cook undisturbed until lightly browned, it will take about 3 to 5 minutes.
Keep stir and fry, breaking up any clumps with the back of a spoon and scraping up any browned bits from the pan until the meat is lightly browned all over 6 to 8 minutes.
Once the beef is cooked well, transfer it to a large bowl.
In the same frying pan, reduce the heat to medium and heat the remaining 1 tablespoon of olive oil, and add the reserved chopped peppers, onions, garlic, thyme, 4 teaspoons salt, and 1/2 teaspoon pepper.
Cook and stir occasionally, until tender but not browned, about 10 to 12 minutes.
Add the tomatoes and keep cooking and stir occasionally, until falling apart and much of the liquid has evaporated, it takes about 10 minutes.
Add the tomato paste and cook, stirring constantly until brick red-colored, about 2 minutes.
Stir in the vinegar and cook until the mixture is reduced, very thick, and if you will use wine so until no smell of alcohol remains 6 to 8 minutes.
Add the broth and bring to a boil then transfer to the bowl with the beef.
Stir in the rice until completely combined. (The mixture will be a little bit wet.)
Now, season with salt and pepper.
Preheat the oven to 450 degrees F.
Place the peppers cut-side up in a baking pan then put the filling into the peppers, gently pressing it in with the back of a spoon.
Make sure you don’t overstuff and split the sides of the peppers. Bake for 15 minutes, then top with the cheese and continue baking until the filling is heated through and the cheese is melted and browned in spots, about 10 to 12 minutes.
Serve it hot with your favorite side dish.