In a large skillet, melt 1 tablespoon of butter over medium heat. Add the diced Vidalia onions and sauté until they become translucent and fragrant, about 5-7 minutes. Season with salt and pepper to taste.
In a medium saucepan, combine the long grain white rice, chicken broth, bay leaves, and a pinch of salt and pepper. Bring to a boil, then reduce heat to low, cover, and simmer for about 18-20 minutes, or until the rice is tender and the liquid is absorbed. Remove from heat and let it sit for a few minutes before fluffing with a fork. Discard the bay leaves.
Lay out the flour tortillas on a clean surface. In the center of each tortilla, layer a generous scoop of the sautéed onions, followed by a portion of the cooked rice, black beans, and a sprinkle of grated cheddar cheese. Feel free to add any other toppings you desire, such as avocado, diced tomatoes, or cilantro.
Fold in the sides of the tortilla, then roll from the bottom to the top, ensuring the filling is enclosed. Heat a skillet or panini press with 2 tablespoons of butter and place the burritos seam-side down. Cook until golden brown and crispy.