In a large cast-iron skillet, heat the oil on medium-high heat until shimmering. Add the ground beef, and cook, stirring occasionally, until browned.
Add the garlic, onion and Worcestershire sauce. Cook until the onion is softened. Remove from heat, stir in the ketchup and mustard, and season with salt and pepper to taste. Set beef mixture aside to cool completely. Meanwhile, heat the oven to 400°F.
Spoon half of the beef mixture into the lined pan, then shower with 1/2 of the cheddar. Tear 2 of the American cheese slices up, and scatter on top. Repeat layering with the remaining beef, cheddar, and American cheese.
Lay the 2nd puff pastry sheet over the filling, and pinch and crimp as desired to seal the pie.
Brush the pie lightly with the egg mixture, and sprinkle with the sesame seeds.
Bake until nicely golden brown, about 45 minutes, tenting with foil if browning too quickly. Let rest for 10 minutes before cutting into.
In a small bowl, thin the mayonnaise with a splash of water until the consistency of dressing. Fold in the lettuce and tomato, and season to taste. Serve alongside wedges of the pie.