The generous chunks of lobster and shiitake mushroom in this ragu make it more of a stew than a sauce.
Bring a large pot of water to a boil over high heat. Add lobsters and cook for 6 minutes. Remove from water. Remove meat from tails and claws, leaving claw meat intact; set meat aside, reserving shells and bodies for stock.
Heat oil in a large high-sided skillet over medium-high heat. Add shallots and garlic, and cook until soft and translucent. Add mushrooms, celery root, carrot, and celery; cook, stirring, until tender, about 10 minutes. Add Brussels sprout leaves and cook, stirring, until caramelized, 3 to 5 minutes.
Carefully add Madeira and deglaze. Add lobster stock, sage, and tomato; reduce heat to a simmer and cook until thickened, 8 to 10 minutes. Add speck and butter; cook, stirring, until butter is melted.
Add lobster meat to skillet; cover and cook until just heated through, 8 to 10 minutes. Do not boil. Remove skillet from heat and drizzle lobster mixture with olive oil; sprinkle with chives and parsley.
Divide the celery root puree evenly between two plates; top with lobster mixture. Set claws in the center of each plate and garnish with radish sprouts; serve immediately.