Preheat the oven grill or broiler on high. Carefully separate the lobster meat from the shell, keeping the shell intact. Chop the lobster meat into bite-size chunks.
In a frying pan, melt the butter over medium heat. Add the garlic and sauté for about 30 seconds, or until fragrant. Be careful not to let it burn, as burnt garlic can turn bitter.
Add the flour and stir quickly for about 1 minute to form a roux. Gradually pour in the wine and half of the milk, stirring constantly. Add the remaining milk and continue stirring until the sauce becomes smooth and thick. Don't worry if the sauce looks lumpy at first. It will smooth out as you continue to stir.
Add 1/4 cup of the shredded cheese, the Dijon mustard if using, and salt to the sauce. Stir until the cheese has fully melted, then gently fold the lobster into the Mornay sauce. Since the lobster is already cooked, it only needs about 1 minute to warm in the sauce. Taste and adjust the seasoning with more salt or pepper, if needed.
Spoon the lobster and sauce back into the shells, then sprinkle the remaining cheese evenly over the top. Grill or broil for 3 to 5 minutes, or until the cheese has melted and turned golden brown. Garnish with chopped parsley, if using. Serve immediately with salad, hot chips or crusty bread. Enjoy!