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Preheat the oven to 375 degrees F.
Using a grinder, grind the meat into a bowl, taking care not to place too much pressure on the meat. The meat mixture should be light and fluffy. (If using pre-ground meat, remove from the packages and loosen with a fork.)
In a separate bowl, combine the breadcrumbs, milk and egg, creating a wet paste. Set aside.
Add the salt, pepper, cayenne and paprika to the bowl with the meat (do not mix), followed by the onions, carrots, green peppers and celery. Add the wet mixture and then gently fold to combine. Do not over-mix; the meat should remain cold.
Line a loaf pan with 2 layers of plastic wrap, leaving plenty of overhang. Pack the meatloaf mixture into the pan. Fold the plastic wrap tightly around the meat, ensuring there are no holes for steam to escape. Cover the pan with aluminum foil and bake for 1 hour 30 minutes. Let the meatloaf cool, preferably overnight in the refrigerator (but at least 3 hours).
For the meatloaf sandwiches: Carefully slice the meatloaf into 6- to 7-ounce portions (about 1 inch thick).
Heat a large pan over medium-high heat. Add the oil and heat until you see a slight haze and hear a nice sizzle sound. Sear the meatloaf slices until slightly browned. Flip the slices and repeat. Cover the pan and continue to cook for 2 minutes; this allows the meat to steam. Remove the meatloaf slices from the pan and set aside.
Put the tomato and onion slices in the same pan. Sprinkle the garlic and thyme on the vegetables and cook until softened.
Toss the arugula, olive oil, salt and pepper together in a bowl. Butter the toasted potato rolls and spread with mayonnaise. Place some meatloaf, tomato, onion and seasoned arugula in each roll and serve.