Position a rack in the bottom third of the oven and heat the oven to 325°F.
Spread the pork on paper towels to dry for 10 to 20 minutes before browning. (You can use this time to chop the onion, celery, and carrot). If the meat is very wet, pat it dry.
In a 6-quart Dutch oven or other heavy-duty pot, heat the oil over medium to medium-high heat until shimmering hot. Season about one-third of pork with salt and pepper and arrange it in a single layer in the pot (there should be at least 1/2 inch of space between the pieces). Brown well on at least 4 sides, adjusting the heat as necessary; each batch should take about 10 minutes to brown. Transfer the pork to a large bowl or rimmed baking sheet as it browns and repeat with the rest of the pork, seasoning with salt and pepper before browning. Once all of the pork is browned, remove the pot from the heat to let it cool for a few minutes.
Pour all but 2 Tbs. of the fat from the pot. (If there is not enough, add oil to equal 2 Tbs.) Return the pot to medium heat, then add the onion, celery, and carrot. Season with a pinch of salt and pepper, and cook, stirring often and scraping the bottom of the pot with a wooden spatula, until the vegetables begin to soften, 5 to 6 minutes. Stir in the garlic, ginger, lemongrass, coriander, and chiles, and cook, stirring occasionally, until fragrant, 1 to 2 minutes.
Add 1 cup water, stirring with the wooden spatula to dissolve any browned bits on the bottom of the pot. Raise the heat to medium high and boil to reduce by about half, 5 to 8 minutes. Add the chicken broth and an additional 1-1/2 cups water. Bring to a boil. Return the pork to the pot along with any accumulated juice. Lower the heat to maintain a simmer.
Crumple a 12x16-inch piece of parchment, then flatten it out. (Crumpling makes for easy handling.) Place the parchment directly on the surface of the stew, allowing the ends to come up the sides of the pot. Cover and put in the oven.
After 30 minutes of stewing, add the onions to the pot. Cover with the parchment and lid, return to the oven. After another 30 minutes, add the squash and bell peppers. Cover with the parchment and lid, return the pot to the oven, and cook until the pork is fork-tender, 1/2 to 1 hour more.
Stir in the cilantro, lime juice, and fish sauce. Degrease the stew by laying a clean paper towel over the surface of the stew and gently pushing it into all the bumps and dips, then quickly peeling it off. Repeat as necessary with more paper towels. Season to taste with salt and pepper and serve.
The stew can be made up to 2 days ahead: Skip the degreasing step, cool to room temperature, cover, and refrigerate. Once the stew is chilled, lift the solidified fat off the top with a slotted spoon. Reheat the stew over medium-low heat to serve.