Preheat the oven to 325°F.
Pat the lamb dry with paper towels and season with salt and pepper.
Heat olive oil in a large Dutch oven over medium-high heat. Working in batches, sear the lamb on all sides until browned. Transfer to a plate.
Lower the heat to medium. Add the chopped onion and sauté for about 5 minutes, until softened.
Sprinkle in the flour and stir to coat the onions.
Stir in the garlic and tomato paste. Cook for 1 minute, just until fragrant.
Pour in the broth, and stir well to combine. Then add the carrots, parsnips, potatoes, mushrooms, thyme, rosemary, and bay leaves. Stir it gently.
Return the lamb to the pot and bring everything to a gentle simmer.
Cover with the lid and transfer to the oven.
Braise for 2 to 2½ hours, or until the lamb is fork-tender and the vegetables are fully cooked.
Remove the bay leaves, taste, and adjust seasoning if needed.