Start by preheating your oven and lining a baking sheet. Combine the dry ingredients, then cut in the cold butter and shortening until the mixture looks crumbly. Add the egg, buttermilk, and cheddar cheese, mixing gently until a soft dough forms.
Pat the dough out on a floured surface and cut into thick rounds or squares. Place them on the baking sheet and sprinkle extra cheese on top. Bake until the biscuits rise and turn golden brown with crisp edges.
Steam the lobster until fully cooked, then let it cool slightly before removing the meat. Chop the lobster into bite-sized pieces and toss with melted butter and chopped chives for rich flavor.
Slice the warm biscuits in half and generously fill them with the buttery lobster mixture. Serve immediately while everything is warm and fresh.