On a work surface, make sandwiches using 2 of the chicken breast cutlets for each: layer 1 of the cutlets, a slice of Swiss cheese and a slice of ham, then another slice of cheese and a slice of chicken. Fasten the sandwiches in the center with a toothpick.
On a plate, mix the 1/2 cup of flour with the paprika and season with 1/2 teaspoon of salt and 1/4 teaspoon of pepper. Dredge a sandwich in the flour to coat lightly on both sides, shaking off any excess. Repeat with the remaining sandwiches.
In a 12-inch skillet, melt 2 tablespoons of the butter in the olive oil and cook the sandwiches over medium-high heat until cooked through and golden, turning once, about 3 minutes per side.
Meanwhile, in a medium saucepan, melt the remaining 2 tablespoons of butter over medium heat. Whisk in the remaining 2 tablespoons of flour until smooth. Whisk in the vermouth and cook for 1 minute (optional). Slowly whisk in the heavy cream and cook, stirring continuously, until the sauce comes to a soft boil and thickens, about 5 minutes. Stir in the torn up Swiss cheese and cook over low heat, stirring continuously, until the cheese is melted, 2 to 3 minutes. Season the sauce with salt and pepper and serve the fondue-style sauce in small bowls alongside the chicken.