The night before serving, drizzle steak evenly with lime juice and 1 tablespoon olive oil. Combine shawarma spices and sprinkle evenly over both sides of steak. Cover and refrigerate overnight or at least 12 hours.
When ready to eat, bring steak to room temperature.
In a food processor, pulse all chimichurri ingredients until combined. Add more olive oil or lemon juice if you want it less thick and/or more tart. Set aside.
To make steak, heat a large saute pan over medium-high heat. Add 1 tablespoon olive oil and heat. Cook steak 2-3 minutes per side until charred and juices just start to form for medium rare. Set aside to let rest.
Whisk eggs together in a small bowl with salt and pepper until no whites are showing. Heat a medium non-stick pan over medium heat. Add olive oil and lower to medium-low heat. Add eggs and let set for a moment. The using a spatula, push the eggs towards the center. Let set and repeat. You want to cook the eggs over low heat so they don't dry out. Low and slow is key.
Serve chimichurri over steak with eggs and top the whole thing with a squeeze of lime and freshly cracked pepper. EAT!