Remove stems from mushrooms and tear into 1-in pieces (leave smaller ones whole) Heat 1 tablespoon olive oil in a dutch oven or large saucepan over medium-high heat until shimmery. Working in batches, add half of mushrooms and saute several minutes until tender. Repeat with remaining mushrooms and 1 tablespoon olive oil.
Heat remaining 1 tablespoon olive oil in same pan over medium-high heat until shimmery. Add carrot and shallot and saute 3-5 minutes until lightly browned. Add garlic and tomato paste and cook several minutes until fragrant.
Pour wine into pan and use a wooden spoon to scrape up any browned bits. Bring wine to a simmer over medium-high heat and cook until reduced in half.
Add pearl onions, broth, bay leaf, thyme, rosemary, and mushrooms to pan. Bring to a simmer over medium-high heat, then reduce heat to medium-low, cover, and let simmer 20-25 minutes until vegetables are very tender and liquid is reduced.
In a small bowl, whisk flour with a little braising liquid from pan until smooth. Fold into bourguignon until smooth and thickened. Discard bay leaf, thyme, and rosemary sprigs and keep bourguignon warm until ready to serve.