Cut the beef chuck roast into four pieces. Heat 1 tablespoon of olive oil in a large, heavy-bottomed pot over medium-high heat until it shimmers. Add the beef chunks and sear until browned on all sides, about 8 to 10 minutes. Remove and place on a plate.

Lower the heat to medium. Add 1 tablespoon of olive oil and the Italian sausage links to the pot. Cook, turning occasionally, until browned all over, about 5 minutes. Transfer the sausage to the plate with the beef.

Remove the pot from the heat and let it cool for about 2 minutes. Meanwhile, finely chop the garlic and remove the leaves from the basil sprigs.

Return the pot to medium heat, add the remaining 2 tablespoons of olive oil, and heat until shimmering. Add the garlic and cook, stirring frequently, until it turns light golden-brown, about 1 minute. Stir in the tomato paste and cook, stirring constantly, until the paste deepens in color, about 1 minute.

Add the whole peeled tomatoes with their juices to the pot. Fill one of the empty tomato cans halfway with water and add it to the pot. Break up the tomatoes using a wooden spoon or potato masher. Add the basil leaves, then season with kosher salt, granulated sugar, and black pepper.

Increase the heat to medium-high and bring the sauce to a rapid simmer. Return the beef and sausages, along with any juices, to the pot. Lower the heat to low and partially cover. Simmer gently, stirring occasionally, until the beef is tender and shreds easily, 3 1/2 to 4 hours.

Remove the beef and sausages to a cutting board. Shred the beef into bite-sized pieces and slice the sausages thickly on a diagonal. Return both to the sauce. Alternatively, serve the meat whole on a platter alongside the sauce and pasta.

Adjust the seasoning with additional salt, pepper, and granulated sugar if needed. Serve the sauce over cooked spaghetti.