In a large nonstick skillet, melt butter over medium heat. Add onions, salt and pepper and saute 5-8 minutes or until onions are soft and have browned. You really want to brown and soften the onions to get the most flavor out of them. Remove from heat and set aside.
Cook the Rice
In a small pot, melt butter over medium heat. Add rice, salt and pepper and toast rice slightly. Add in chicken broth and bay leaves. Cover, bring to boil and then reduce heat to simmer until rice is completely cooked, about 20 minutes. Remove bay leaves, fluff rice with fork and set aside.
Make the Burritos
Heat another skillet or panini press to medium heat. Steam or microwave tortillas so they are soft and easy to work with. Build your burritos by adding 1/4 cup grated cheese to the center of the tortilla. On top of the cheese, add beans, rice, cooked onions and any other desired toppings. Top your mountain of fillings with another 1/4 cup of cheese and fold tightly into a burrito, tucking the edges as you go.
Brush hot panini press with melted butter or melt butter into the bottom of preheated skillet. Place burritos seam side down and cook until golden in color and crispy. Flip to brown opposite side. Remove from skillet and serve immediately with any desired toppings.