Heat a large skillet or griddle to medium heat. Spread butter one side of the bread and place butter side down onto the griddle. Place 1 slice of cheese on top of the bread. Layer 2 slices of turkey on top of the cheese and top with an additional slice of cheese.
Add the other bread slice and butter the top. Cook until golden brown on the bottom, flip and cook until golden brown on the other side. Repeat with remaining sandwiches.
Melt the butter in a skillet over medium-high heat. Whisk in the flour and cook for 2-3 minutes, be careful not to burn the butter. Add the chicken broth a little at a time, while whisking. Cook until the gravy has thickened. Season with pepper.