Brush both sides of the sourdough slices with enough olive oil to coat the bread. Broil for 2 to 3 minutes on each side, until golden brown on both sides.
Cut the garlic cloves in half and rub each bread slice on both sides with the cut side of the cloves.
Top the garlic-rubbed slices of bread with 1 tablespoon each of the grated Parmesan cheese, then broil until the cheese melts, about 1 minute.
Place the bread on a cutting board and cut into 1-inch cubes.
Slice the red onion and avocado and set aside.
Zest and juice the lemon into a small bowl. Add 2 tablespoons of the thinly sliced chives and the remaining 2 tablespoons of the grated Parmesan. Season with salt and pepper. Slowly drizzle 1/4 cup of the olive oil into the dressing, whisking continuously until combined.
In a large bowl, add the lettuce, avocado, red onion and sourdough garlic bread croutons. Season with salt and pepper, add the dressing and toss with your clean hands--they're the best tools.
Plate the salad, sprinkle with the remaining sliced chives and enjoy.
For the dressing, you can also add the ingredients to a Mason jar, cover and shake until combined.