Put the rice, a big pinch of salt, and 1 1/2 cups of water in a 3-quart saucepan. Bring to a boil over medium-high heat, reduce the heat to low, cover, and cook until the rice has absorbed the water and is tender, about 15 minutes. Remove from the heat and set aside with the lid on.
Meanwhile, heat 1 tbsp of the oil in a 4-quart saucepan over medium heat. Add the onion, bell pepper, garlic, and a pinch of salt; cook, stirring occasionally, until softened about 3 minutes. Add the cumin and cook until fragrant, about 30 seconds.
Add the tomato sauce and stir for 1 minute. Add the beans and 1 cup of water and simmer until the liquid reduces to the level of the beans, about 4 minutes.
Add the rice to the beans and mix well. Stir in the Salsa Lizano and season to taste with salt and pepper. Keep warm.
Heat the remaining 1 Tbsp of oil in a 12-inch nonstick skillet over medium heat, swirling the pan to coat evenly. Gently crack the eggs into the pan. Season with salt and pepper, cover, and cook until the yolks’ edges have just begun to set, 2 to 3 minutes. (The eggs should cook gently, so lower the heat if needed.) Separate the eggs with the edge of a spatula.
To serve, put a heaping spoonful of the rice and beans on a plate and slide 2 eggs on top. Sprinkle with the cilantro.