For the strip steak sandwiches: Heat 2 grill pans over high heat.
Brush the onion slices on one side with oil and place them oiled-side down on the grill pans. Cook until grill marked, 3 to 4 minutes, then brush the tops with more oil, flip the slices and cook until grill marked on the other side, another 3 to 4 minutes. Set aside.
Season the strip steaks on both sides with the lemon pepper, salt and black pepper. Grill for 2 minutes, flip and grill an additional 2 minutes for rare.
Grill the rolls cut-side down until toasted, about 2 minutes.
For the eggs: Heat a large nonstick skillet over medium heat and add the butter. Add all the eggs (or cook in batches if needed), sprinkle with salt and pepper and fry for 2 minutes. Flip the eggs and cook for 45 seconds.
For the ancho mayo: Mix together the mayonnaise and chile powder in a small bowl
Slather both sides of the rolls with some ancho mayo and top with a steak, some onions and an egg.