Preheat oven to 400 degrees F or 204 degrees C

Peel ½ head of garlic and generously coat with olive oil. Bake for 10 minutes and remove from oven, set aside.

Cover potatoes with water in large saucepan. Add 1 tablespoon salt, bring to simmer over medium-high heat, and cook until potatoes are tender, 8 to 10 minutes.

Drain potatoes and transfer to large bowl for mashing. Add in your roasted garlic. Using potato masher, mash potatoes until smooth. Whisk parmesan cheese, milk and egg together, then stir into potatoes along with 6 tablespoons melted butter. Season with 1 teaspoon salt, and ½ teaspoon pepper; cover and set aside.

Melt 2 tablespoons butter in skillet. Add carrots, celery onion, and ¾ teaspoon salt and cook until soft and tender. Add ground beef and cook, breaking up meat with wooden spoon, until no longer pink, about 10 minutes.

Stir in tomato paste, garlic, and thyme. Stir in flour and cook for 1 minute. Slowly stir in broth and Worcestershire. Bring to simmer and cook, stirring occasionally, until mixture has thickened slightly, about 10 minutes.

Turn off heat and add in peas and carrots.

Smooth mashed potatoes over ground beef filling. If desired, use a fork to make ridges on the surface.

Bake for 25 minutes.

Place skillet in broiler for 2-3 minutes to allow it to get golden brown and crusty.