1. Spread one tablespoon pesto onto one side of each bread slice. Divide chicken strips among 4 slices of bread. Place a slice of cheese on top of chicken. Top with remaining bread slices, pesto side down.
2. Brush the outer sides of each sandwich with olive oil.
3. Heat large skillet or griddle over medium. Grill sandwiches until browned on one side; turn and grill until brown on other side.
Serving Suggestion: Cut sandwiches diagonally. Serve with pasta salad. Refrigerate leftovers.