Position a rack in the bottom third of the oven and heat the oven to 325°F.
Spread the beef 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, cook the pancetta in the oil over medium heat, stirring occasionally, until browned but not crisp, 6 to 8 minutes. Using a slotted spoon, transfer to a paper-towel-lined plate and set aside. Do not wipe out the pan.
Heat the pancetta fat over medium to medium-high heat until shimmering hot. Season about one-third of the beef 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 beef to a large bowl or rimmed baking sheet as it browns and repeat with the rest of the beef, seasoning with salt and pepper before browning. Once all of the beef 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 yellow 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, tomato paste, soaked porcini, rosemary, and bay leaf and cook, stirring occasionally, until fragrant, 1 to 2 minutes. Return the pancetta to the pot.
Add the wine, 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 reserved mushroom soaking liquid. Bring to a boil.
Return the beef 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 1 hour of stewing, add the pearl onions to the pot. Cover with the parchment and lid, return to the oven. After another 30 minutes, add the tomatoes. Cover with the parchment and lid, return the pot to the oven, and cook until the beef is fork-tender.
Stir in the basil. 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.