Heat 1 tablespoon olive oil in a medium cast-iron skillet over medium-high heat. Add the green garlic, season with salt and pepper, and cook for 5 minutes or until soft and lightly browned, stirring occasionally. Add the thyme and lemon zest, and cook for an additional 2 minutes, stirring. Remove from the heat and set aside in a warm place.

Generously season the steak with salt and pepper. Heat a cast-iron grill pan over high heat. Add the steak to the skillet and sear on all sides, approximately 7-8 minutes total. Remove from the heat and transfer to a cutting board. Cut the steak diagonally into thick slices. Reheat the skillet over medium-high heat and add 1/2 tablespoon olive oil. Return the steak slices to the skillet and cook for 1 minute. Add the lemon juice and cook for an additional 2 minutes, turning, or until desired doneness. Remove from heat and set aside.

Poach the quail eggs.

Transfer the steak to a serving platter or individual plates. Arrange the quail eggs on top of the steak. Sprinkle with additional thyme, if desired. Serve with the green garlic on the side. Serve immediately.