Preheat an outdoor grill or indoor grill pan over medium heat. In a pan, heat 1 tbsp. EVOO and the butter over medium-low heat until the butter foams. Stir in half of the anchovy paste and garlic and cook for 2 minutes; remove from the heat.
In a small bowl, combine the remaining anchovy paste and garlic, the lemon juice, worcestershire, mustard and egg yolk, if using. Whisk in about 1/3 cup cheese and lots of black pepper.
Brush the cut sides of the romaine with EVOO to lightly coat. Grill the romaine until charred, 4 to 5 minutes.
Grill the meat, turning once and basting with the anchovy butter, for 6 to 8 minutes. Transfer to a cutting board and let rest for 10 minutes. Slice.
Top the romaine with dressing, pita chips and cheese. Serve with the steak.