Warm turkey with broth in a covered skillet over low heat for 5 to 8 minutes until the meat reaches 165°F. Uncover and let excess liquid evaporate, then transfer to a bowl.
Divide half the Muenster over 2 bread slices. Add warm turkey in flat layers, scatter the finely chopped drained giardiniera evenly and cover with remaining cheese and bread.
Spread softened butter thinly over both outer faces of each sandwich. Wipe the skillet and heat over medium-low.
Cook for 4 to 5 minutes per side until golden and cheese is fully melted, covering briefly if needed while keeping the heat low enough to avoid burning the bread.
Rest for 2 minutes, cut each sandwich in half and serve while the filling remains soft.