Preheat the oven to 350 degrees F (175 degrees C). Lightly grease a 2-quart baking dish.

Melt butter in a medium pot over low heat. Sauté lobster meat in butter until it starts to change color. Do not cook too long or too fast, or lobster meat will toughen. Use a slotted spoon to remove lobster meat from the pan and set aside.

Stir flour, dry mustard, salt, and pepper into remaining butter in the pot. Gradually whisk in cream and milk. Cook, stirring constantly, until thickened, about 5 minutes.

Return lobster meat to the pot. Tear bread into small pieces and stir into lobster mixture. Pour into the prepared baking dish.

Bake in the preheated oven until bubbly and delicately browned, 20 to 25 minutes.