To make the rosemary-ale jus, melt 1 tablespoon butter in a medium sized saucepan, over medium-low heat; whisk in beef broth, beer, garlic, rosemary, onion powder and 1/4 teaspoon pepper, and allow to simmer.
In a large cast iron skillet, melt 2 tablespoons butter over medium-high heat. Stir in onion, apple, salt and the additional 1/4 teaspoon pepper, and cook for 7 to 8 minutes, stirring frequently, until light golden brown and tender; remove from skillet and set aside.
Divide pastrami equally between 4 slices of bread; top with apples and onions, and white Cheddar cheese. Top with remaining bread slices.
Melt 1 tablespoon butter in same skillet; place 2 sandwiches in pan, place the bottom of a heavy skillet on top, and press down to flatten. Cook for 2 minutes, flip, press down with heavy skillet again, and cook an additional 2 minutes, or until crisp. Repeat process with remaining 1 tablespoon butter and additional 2 sandwiches; slice sandwiches in half, diagonally. Divide rosemary-ale jus into 4 small bowls, and serve with panini for dipping.
A viewer or guest of the show, who may not be a professional cook, provided this recipe. It has not been tested for home use.