In a large bowl, add the chuck roast, short ribs, garlic, calamansi juice, soy sauce, and ground pepper, and thoroughly mix until coated. Cover and refrigerate for 4 to 24 hours.
In a large 10 to 12-quart pot over low to medium heat, add the pork belly and cook until golden brown and crispy, about 6 to 8 minutes. Using a slotted spoon, remove the pork and set it aside. There should be plenty of rendered fat in the pot.
Turn the heat up to high, and once it begins to smoke lightly, add the carrots and potatoes and sauté for 6 to 7 minutes, or until they start to brown. At this stage, add in the pepper and saute for a further 2 to 3 minutes. Season with salt and stir to combine, then set the vegetables aside and refrigerate.
In the same pot, add the butter over medium heat. Once melted, add in the onions, give them a gentle season of salt, and saute for 5 minutes or until lightly browned. Then add in the brown sugar and stir to combine. Cook for a further 10 to 15 minutes or until well caramelized.
Stir in the garlic and cook just until fragrant, which takes 30 to 45 seconds. Set the onions and garlic aside on a separate plate or platter.
Add the oil to the pot and turn the heat to high. Once it begins to smoke lightly, add in the chuck roast and short ribs and sear for 2 to 3 minutes per side or until well browned on all sides. You may need to do this in batches. Set it aside with the onions.
In the empty pot over medium heat, add the tomato paste, stir, and cook it for 3 to 4 minutes or until it turns a rust-like color. Then add in the tomatoes, along with any marinade liquid, to the pot. It should almost immediately thicken.
Next, add in the fish sauce, beef stock, vinegar, bay leaves, salt, and pepper. Taste it, it should be delicious. Stir to combine, then add back the seared beef, crispy pork belly, and caramelized onions and garlic.
Cover and cook over low to low-medium heat for 2 ½ to 3 hours or until fork-tender.
Next, stir in the sauteed carrots, potatoes, and peppers and cook for an additional 20 minutes or until tender.
Finish with lemon juice and adjust any seasonings.