Heat oven to 450 degrees. Cut away the flat bottom of the sea-urchin shell with scissors; pour away liquid. Scrape out membrane with the end of a spoon's handle. Set aside the six most attractive urchins in a baking dish.
Scoop the flesh out of the remaining six to yield about 1/2 cup. Pulse in food processor for 5 to 8 seconds. Add butter, and blend until thoroughly emulsified; add more butter if necessary. Press the urchin-butter mixture through a sieve with a spatula.
Bake the whole sea urchins 4 to 5 minutes.
Meanwhile, combine stock, cream, cayenne, salt, and pepper, to taste, in a small saucepan over medium heat. Add urchin-butter mixture, stirring constantly, until resulting emulsion thickens. Adjust seasoning; add lemon juice.
Remove sea urchins from oven. Arrange them on a bed of seaweed, drizzle with Sea Urchin Butter, and garnish with lemon halves.