Heat 2 tablespoons of olive oil in a large Dutch oven over medium-high heat. Sear beef chuck cubes until browned, about 3-4 minutes per batch. Remove and set aside.

In the same pot, add chopped onions and cook until golden, about 5-7 minutes. Stir in minced garlic and cook for 1 minute.

Add sliced carrots and cook for 2-3 minutes. Stir in tomato paste and cook for 1 minute.

Sprinkle flour over the vegetables, stir, and add beef stock and water. Return seared beef to the pot, mixing well.

Add thyme, bay leaf, salt, and pepper. Bring to a gentle boil, cover, and simmer on low for 2½ to 3 hours.

In a separate skillet, heat 1 tablespoon of olive oil. Cook sliced mushrooms until golden brown, then add pearl onions and cook until soft.

Fold sautéed mushrooms and pearl onions into the stew. Simmer for another 10 minutes before serving.