Combine mayonnaise, anchovies, cheese, vinegar, mustard, and garlic in a food processor; process until smooth, about 45 seconds, stopping to scrape down sides as needed. Transfer mixture to a bowl; season generously with salt and pepper. Refrigerate, uncovered, until ready to use.
Heat olive oil in a medium skillet over medium until shimmering, about 2 minutes and 30 seconds. Add panko; cook, stirring occasionally, until lightly browned, about 5 minutes. Season with salt and pepper. Transfer to a small baking sheet; let cool 10 minutes. Stir in chives and lemon zest.
Using a small offset spatula, spread anchovy dressing over cut sides of lettuce. Dredge in panko mixture. Transfer to a platter or board, and serve.
Store dressing in an airtight container in refrigerator up to 5 days.
