Place 3 pounds of beef tongue (dila ng baka) in a deep pot and cover with water. Bring to a rolling boil (100°C/212°F) for 10-15 minutes. Drain and rinse the tongue thoroughly, then clean the pot for reuse.
Return the tongue to the clean pot and add fresh water to cover. Add 1 small quartered onion, 1 head crushed garlic, 2 bay leaves (dahon ng laurel), 1 teaspoon peppercorns (paminta), and 2 tablespoons salt. Bring to a boil, then reduce to low heat (90°C/195°F). Simmer covered for 4-5 hours until fork-tender, adding water as needed.
Remove tongue from broth, reserving 1 cup of the cooking liquid. Let cool slightly until manageable, then peel the skin while still warm. Refrigerate for 1-2 hours until firm, then slice diagonally into ½-inch pieces.
Heat ¼ cup canola oil in a large skillet over medium heat (175°C/350°F). Fry the sliced potatoes until golden brown; set aside. In the same pan with 2 tablespoons oil, sauté mushrooms for 1 minute and set aside. Cook chopped onions and garlic until soft, then brown the sliced tongue for 3-4 minutes.
Add 1 cup tomato sauce, the reserved broth, and 2 tablespoons soy sauce. Bring to a boil, then reduce heat to low (140°C/285°F). Simmer covered for 10-15 minutes. Add 1 tablespoon sugar, salt, and pepper to taste. Return potatoes and mushrooms to the pan.
Cook until the sauce thickens enough to coat the back of a spoon. Let rest for 5-10 minutes before serving. The internal temperature of the meat should reach 74°C/165°F when reheating.