Step 1

To make the giardiniera, combine all of the ingredients in a saucepan and bring to a boil over high heat, stirring to dissolve the salt. Transfer to a heatproof bowl, let cool, and then let marinate in the liquid until serving. (The giardiniera will keep in an airtight container in the refrigerator for up to 1 week.)

Step 2

To make the relish, combine all of the ingredients in a food processor and pulse until finely chopped and incorporated.

Step 3

To cook the hot dogs, in a saucepan, combine the beer and hot dogs and bring to a boil over high heat. Remove from the heat and let the hot dogs cool in the beer. On a charcoal or gas grill or a stove-top grill pan, sear the hot dogs over high heat, turning as needed, until lightly charred.

Step 4

Line up half of the bread slices on a work surface. Top each slice with 1/8 of the cheese, 3 tablespoons pickle relish, 1/2 hot dog (2 pieces), 1/4 cup drained giardiniera, 2 tomato slices, and another 1/8 of the cheese. Close the sandwiches with the remaining bread slices.

Step 5

Line a large platter with paper towels. In a skillet over high heat, melt 1 tablespoon of the butter. Turn down the heat to low, add 1 sandwich, and cook, turning once, for 2 to 3 minutes on each side, until browned and crisp on both sides and the cheese is melted. Transfer to the prepared platter to blot the excess grease. Repeat with the remaining butter and sandwiches.

Step 6

Cut the sandwiches in half, plate, and serve.