To make the soup, season the pork all over with the salt, cumin and pepper. In a 5-quart (5-l) Dutch oven or other large pot over medium-high heat, heat the olive oil until hot and shimmering. Working in batches to avoid overcrowding (which would cause the pork to steam rather than brown), sear the pork until browned all over, 3 to 4 minutes per side. Transfer the pork to a plate.
Reduce the heat to medium-low, return all the pork back to the pot and add the onion, garlic and bay leaves. Pour in the chicken broth, cover and cook, stirring every 30 minutes, until the pork is tender and easily breaks apart, about 1 1/2 hours. While cooking, add additional water, 1 cup (8 fl. oz./250 ml) at a time, as needed while the broth reduces, making sure the meat is always submerged in the liquid.
Meanwhile, make the salsa verde: Preheat an oven to 400°F (200°C). Line a baking sheet with aluminum foil.
Place the tomatillos, poblanos, jalapeños, bell pepper, onion and garlic on the prepared baking sheet. Drizzle with the olive oil and sprinkle with 1/8 tsp. of the salt. Toss to coat. Roast until the vegetables are charred, 20 to 25 minutes. Allow the roasted vegetables to slightly cool, about 10 minutes.
In a blender, combine the cooled vegetables, cilantro, chicken bouillon powder, oregano, onion powder, garlic powder and remaining 1 tsp. Scoop out 1/2 (4 oz./125 ml) of the from the and add to the blender. Blend until smooth, cracking the lid away from you to let the steam escape.