Melt the butter in a saucepan over medium heat. Add the garlic and stir in 1/2 cup flour. Cook, stirring constantly, for about 2 minutes.
Whisk in the milk, and cook, whisking constantly, until the sauce thickens.
Bring to a boil, then add the cheeses and stir until melted and smooth. Add cayenne, salt and pepper to taste.
Remove from the heat and pour into a 9-inch (23-centimeter) square pan. Cover with plastic wrap. Refrigerate until firm, at least 3 hours.
Remove the plastic wrap and invert the cheese mixture onto a work surface. Cut into 6 large or 12 medium squares.
Put the remaining 1/2 cup flour, the eggs and breadcrumbs in 3 separate bowls.
Heat the oil in a deep fryer to 350 degrees F (180 degrees C).
Dredge each cheese square in the flour, then dip in the eggs and finally coat with the breadcrumbs. Fry until golden brown, about 2 minutes. Drain on paper towels.
To serve, place 1 large or 2 medium fried cheese squares on each plate. Toss the arugula with lemon juice, olive oil and salt and pepper to taste. Top the fried cheese with pickled vegetables and the arugula.
You can substitute any of the cheeses in this recipe with your favorites. The cheese squares can be frozen before frying to make them easier to work with.