Cook potatoes: Peel and dice potatoes into ½-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.
Mash potatoes: Strain potatoes and return to the pot. Mash the potatoes until smooth. Stir in half-and-half, butter, 1 tsp. kosher salt, minced garlic, and chives. Once fully combined, stir in the two egg yolks until completely incorporated, and set aside.
Cook vegetables: 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.
Cook ground beef: Add the ground beef, and stir until the meat is fully cooked. Add ½ tsp. kosher salt, tomato sauce, breadcrumbs, Worcestershire sauce, and tomato paste.
Assemble casserole: Evenly spread the meat filling into a 2 ½-qt. casserole dish. Top the meat mixture with the mashed potatoes, carefully spreading the mashed potatoes over the entire surface of the meat.
Bake casserole: Bake for 25 minutes or until potatoes are lightly browned.
