Bring broth to a gentle simmer in a small saucepan. Add chicken cutlets and poach gently for 8 to 12 minutes until the thickest part reaches 165°F.
Lift chicken out, drain and chop into small pieces. Discard the poaching broth, then warm hot sauce and sauce butter in the empty pan and toss the chicken through them.
Divide half the provolone over 2 bread slices, add the warm chicken and blue cheese, then top with remaining provolone and bread.
Butter the outside faces. Cook both sandwiches in a large skillet over medium-low heat for 4 to 5 minutes per side, covering for 2 minutes after turning, until golden and the filling reaches 165°F.
Rest for 1 minute and cut in half before serving.