Cut 2 pieces of cheesecloth into 12-by-22-inch rectangles; lay them on a clean work surface, overlapping each other perpendicularly in the center to form a cross. Pile onion, carrots, garlic, parsley, thyme, rosemary, bay leaves, and peppercorns in center. Gather ends together to enclose contents completely, and tie the top with kitchen twine. Set aside.
Preheat oven 300 degrees. Heat 2 tablespoons oil in a large Dutch oven over medium heat. Add salt pork, and saute until brown and crisp, about 7 minutes. Using a slotted spoon, transfer pork to baking sheet, leaving rendered fat in Dutch oven.
Season beef cubes with salt and pepper. Working in 2 batches, place beef in Dutch oven in a single layer; cook until dark brown on all sides, about 8 minutes total. Transfer to baking sheet with salt pork before adding next batch.
Using tongs, transfer beef and pork to Dutch oven. Sprinkle flour over and toss to coat. Slowly whisk in beef stock and bring to a simmer over medium heat, stirring frequently until thickened, about 8 minutes.
Add wine and tomato paste and season with salt and pepper; stir to combine. Add cheesecloth bundle and bring to a boil over high heat. Cover; transfer to oven. Cook until beef is very tender, about 2 1/2 hours.
Remove pot from oven and transfer cheesecloth bundle to a large sieve set over a bowl. With wooden spoon, press on bundle to release as much liquid as possible. Discard bundle and pour accumulated juices into Dutch oven.
Remove beef and pork from Dutch oven. Boil over high heat until it is reduced to about 4 cups. Skim surface as needed with large metal spoon. Reduce heat to low; return beef and pork to Dutch oven.
While sauce is reducing further, set a large skillet over high heat; add butter and remaining tablespoon oil. Add pearl onions, water, sugar, and a large pinch of salt. Bring to a boil and reduce heat to medium; simmer until almost all the liquid evaporates, 6 minutes. Raise heat to medium-high and add mushrooms. Cook, stirring occasionally, until vegetables are browned and glazed, 4 to 5 minutes. Remove from heat. Transfer mushrooms and pearl onions to Dutch oven and simmer over medium heat until heated through. Adjust seasoning with salt and pepper as desired, and serve at once. Garnish each serving with chopped parsley.