Preheat a large grill pan over medium-high heat. Brush the bread slices with 1 tablespoon EVOO and grill, turning once, until grill marks appear, 1 to 2 minutes; rub each crostini with the halved garlic.
Rub the steak with 1 tablespoon EVOO and season with salt and pepper; grill, turning once, until an instant-read thermometer registers 135 degrees for medium rare, 3 to 5 minutes per side. Let rest, then thinly slice against the grain and on an angle.
Meanwhile, fill a large, deep skillet halfway with water and bring to a simmer over medium heat; add 1 tablespoon vinegar. Working with 1 egg at a time, crack into a small bowl and gently slide into the water. Cook until the whites are opaque but the yolks are still runny, 3 to 5 minutes. Using a slotted spoon, gently transfer the eggs to a kitchen towel to drain.
In a salad bowl, whisk together the grated garlic, remaining 1/4 cup vinegar and the mustard; slowly whisk in the remaining 1/2 cup EVOO and season with salt and pepper. Add the mixed greens, arugula and 1/4 cup parmesan and toss. Divide among 4 serving plates and top each with some steak and a poached egg. Sprinkle with the remaining parmesan. Serve with the grilled garlic crostini.