Place 1/4 cup of the flour and the measured salt and pepper in a large bowl and whisk to combine; set aside. Trim the roast of excess fat and sinew and cut it into 1- to 1-1/2-inch cubes. Place the meat in the flour mixture and toss to coat; set aside.
Heat the oil in a large, heavy-bottomed pot or Dutch oven over medium heat until shimmering. Shake off the excess flour from about one-third of the meat and add it to the pot. Cook, stirring rarely, until browned all over, about 4 to 5 minutes. Remove to a large bowl. Repeat with the remaining meat in 2 more batches; set aside.
Add the onion to the pot and season with salt and pepper. Cook, stirring occasionally, until softened and just starting to brown, about 5 minutes. Add the tomato paste, stir to coat the onion, and cook until the raw flavor has cooked off, about 1 to 2 minutes.
Sprinkle in the remaining tablespoon of flour and cook, stirring occasionally, until the raw flavor has cooked off, about 1 minute. Pour in the wine, scrape up any browned bits from the bottom of the pot, and cook until the mixture has thickened, about 3 minutes.
Return the meat and any accumulated juices in the bowl to the pot. Add the broth, bay leaves, and thyme and stir to combine. Increase the heat to high and bring to a boil. Immediately reduce the heat to low and simmer uncovered for 1 hour.
Cut the carrots, celery, and potatoes into large dice and add them to the pot (peel the carrots and potatoes first, if desired). Stir to combine, cover with a tightfitting lid, and simmer, stirring occasionally, until the vegetables and meat are knife tender, about 1 hour more.
Remove and discard the bay leaves and thyme stems. Stir in the peas and simmer uncovered until warmed through, about 5 minutes. Taste and season with salt and pepper as needed.
Slow Cooker Method
Heat the oil in a large frying pan over medium heat until shimmering. Add the celery, carrot, onion, garlic, and thyme and cook, stirring occasionally, until the carrots just begin to soften, about 10 minutes. Add the tomato paste, stir to coat the vegetables, and cook until itís no longer raw-tasting, about 3 minutes. Add the wine and mustard, stir to combine, and bring to a simmer. Cook until the wine reduces by roughly half, about 5 minutes. Remove from heat and set aside. (At this point the mixture can be completely cooled and refrigerated up to 12 hours until ready to continue.)
Place the flour, measured salt, and measured pepper in a large bowl and whisk to combine; set aside. Trim the roast of excess fat and sinew and cut it into 1- to 1-1/2-inch cubes. Place the meat and mushrooms in the flour mixture and toss to coat. Pour all of the meat, mushrooms, and loose flour mixture into the slow cooker. Add the vegetable-wine mixture and stir to combine. Add the beef broth and bay leaf, tucking the bay leaf into the broth.
Cover and cook on high, stirring about every 1 1/2 hours if possible (if you canít stir it that often, itís OK), until the beef is tender and can easily be pierced with a fork, about 5 hours. Add the peas and parsley, stir to combine, and continue to heat until the peas are warm, about 5 minutes. Taste and season with more salt and pepper as needed.