Peel and dice potatoes into 1/2-inch cubes. Place into medium-sized pot with a lid, and cover potatoes with cold water. Cover the pot and bring to a boil over high heat, then remove the lid of the pot and reduce the heat to medium-low, allowing the potatoes to simmer until tender, about 15 minutes.

Strain potatoes and return to the pot. Mash the potatoes until smooth. Stir in half-and-half, butter, 1 tsp. kosher salt, 1 clove of garlic, and chives. Once fully combined, stir in the two egg yolks until completely incorporated, and set aside.

Preheat the oven to 400°F. Heat the vegetable oil in a large skillet until shimmering. Add onion, bell pepper, and celery. Sauté over medium-high until completely soft, about 8 minutes.

Add the ground beef and stir until the meat is fully cooked. Add 1/2 tsp. kosher salt, tomato sauce, breadcrumbs, Worcestershire sauce, and tomato paste.

Evenly spread the meat filling into a 2 1/2-qt. casserole dish. Top the meat mixture with the mashed potatoes, carefully spreading the mashed potatoes over the entire surface of the meat. Bake for 25 minutes.