Heat a 12-inch cast iron skillet over medium heat.
Spread each bread slice with Roasted Shallot Mayo. Add layers of grated goat cheese, a small handful arugula, and finish with more grated cheese, then top with bread. Repeat for the second sandwich.
Add the sandwiches to the hot skillet and cook until golden brown and crisp and the cheese is melted, 4 to 5 minutes per side. Cut the sandwich in 1/2 before serving.
Preheat the oven to 350 degrees F.
Place the shallot and garlic on a square of aluminum foil. Drizzle lightly with olive oil and sprinkle with salt and pepper. Fold up and seal the foil. Place in the oven and roast for 1 hour. Remove and let cool.
Add the shallot and garlic to a bowl. Add the mayonnaise and mix together until smooth.