Heat a nonstick saute pan over medium-low heat. Add butter and heat until it no longer bubbles. Add shallot, lobster and green asparagus and gently warm through. Meanwhile, crack eggs into a small mixing bowl and whisk until frothy. Incorporate parsley and chives, and season, to taste, with kosher salt and black pepper. Add eggs to saute pan and gently scramble eggs until just barely cooked.
On a 10-inch plate, pour eggs into a 5-inch diameter ring mold and fold the lobster mixture in. Mixture should be very soft. Garnish with toast and oyster cava cream.
In a medium saucepot sweat the shallots and garlic until translucent. Add Cava and reduce to 1 cup. Add the heavy cream, and bring to a low boil. Reduce heat and simmer cream until nap stage (thick enough to coat the back of a spoon). Cool for about 5 minutes. Puree cream in a blender and pass through a chinois. Season with kosher salt, to taste. Store cream refrigerated in a sealed plastic container for up to 3 days.
Just before serving, reheat cream in a small sauce pot, adding additional heavy cream to return sauce to nap stage. Add shucked oysters, including oyster liquor, and gently poach oysters in cream. Add chives and reseason with kosher salt, to taste.