Prepare the Cheese Sauce: In a large saucepan, melt the butter over medium-high heat. Whisk in 1 tablespoon of the flour and cook for 1 minute while whisking constantly. Gradually pour in the milk, whisking continuously to avoid clumps. Bring the mixture to a simmer and cook for about 3 minutes until it is thick enough to coat the back of a spoon.

Combine Cheese and Pasta: Lower the heat and add the shredded white Cheddar, Monterey Jack, salt, garlic powder, black pepper, and cayenne pepper. Stir constantly until the cheese is completely melted and the sauce is smooth. Remove from the heat and gently fold in the cooked elbow macaroni.

Chill the Mixture: Spread the mac and cheese evenly into a lightly greased 13-x 9-inch baking dish. Press a layer of plastic wrap directly onto the surface of the pasta to prevent a skin from forming. Place in the refrigerator and chill for at least 3 hours, or up to 24 hours, until very firm.

Form the Bites: Fill a Dutch oven or high-walled pot with about 2 inches of vegetable oil and heat it to 350°F. While the oil heats, use a 2-tablespoon scoop to portion the chilled macaroni mixture and roll it into 36 uniform balls. Place them on a baking sheet lined with parchment paper.

Coat the Bites: Set up a dredging station with three shallow bowls. Place the remaining 1/3 cup of flour in the first, the whisked eggs and water in the second, and the panko in the third. Dredge each ball in flour, dip it into the egg wash, and then roll it in the panko, pressing gently to ensure the crumbs adhere.

Final Chill and Fry: For the best results, chill the coated bites in the refrigerator for 30 minutes before frying. Once ready, fry the bites in batches of 6 to 8 for about 4 to 5 minutes. Turn them occasionally until they are golden brown and the centers are melty.

Drain and Serve: Use a slotted spoon to transfer the fried bites to a paper towel-lined baking sheet to drain excess oil. Serve them while they are hot and gooey.