In a medium bowl combine mayonnaise, vinegar and garlic. Add chicken and fennel and stir until well combined. Spoon chicken mixture evenly atop 4 of the bread slices, spreading to an even layer atop the bread. Top with arugula, pepper strips and cheese. Top with remaining bread slices.
Lightly coat an unheated panini griddle, covered indoor electric grill or large nonstick skillet with nonstick cooking spray. Preheat over medium heat or according to manufacturer's directions.
Place assembled sandwiches on griddle, grill or skillet, adding in batches if necessary. If using griddle or grill, close lid and grill for 2 to 3 minutes or until bread is toasted. (If using skillet, place a heavy saucepan or skillet on top of sandwiches. Cook for 1 to 2 minutes or until bottoms are toasted. Carefully remove saucepan or top skillet (it may be hot). Turn sandwiches; top again with the saucepan or skillet. Cook for 1 to 2 minutes more or until bread is toasted.)