Make the salad: Light a grill. In a medium bowl, toss the chicken breasts with 2 tablespoons of olive oil, the garlic, salt and pepper. Grill, flipping once, until charred and cooked through, 6 to 7 minutes per side. Transfer to a board and let rest for 5 minutes, then slice into thick pieces.
Meanwhile, brush the romaine and sourdough with the remaining olive oil and season with salt and pepper. Grill the romaine cut-side down until charred, 4 minutes, and sourdough, flipping once, until charred and crisp, 5 minutes.
On a platter, arrange the grilled romaine and top with the grilled chicken. Roughly tear the sourdough into croutons and scatter over top. Drizzle with dressing and garnish with Parmesan, lemon zest and chives, then serve.