In a bowl, combine 1/4 cup soy sauce, 1/4 cup lemon or calamansi juice, half of the chopped onion (about half of the 1 large onion), and all 4 cloves minced garlic. Add the 2 pounds chicken thighs, turn to coat, and marinate in the refrigerator for at least 30 minutes.
Remove the chicken from the marinade and set the marinade aside (do not discard).
Heat 2 tablespoons cooking oil in a large pot over medium heat. Add the remaining chopped onion and sauté until translucent, about 3–4 minutes.
Add the marinated chicken to the pot and brown on all sides, turning as needed, about 5–8 minutes.
Pour the reserved marinade and 1 cup tomato sauce into the pot. Add 2 bay leaves and 1/2 teaspoon black pepper. Bring the mixture to a simmer.
Cover the pot, reduce the heat to medium-low, and simmer the chicken for about 30 minutes, or until the chicken is tender and cooked through.
Add the 1 large cubed potato and the 1/2 red bell pepper and 1/2 green bell pepper (both cubed). Continue to cook, uncovered, until the potatoes are fork-tender, about 10–15 minutes.
Taste and adjust seasoning if necessary. Remove and discard the bay leaves, then serve the chicken asado hot with steamed rice.