Cut the grilling cheese into thick slices, then cut those slices in half.
Assemble a dredging station: Put the flour in a shallow bowl and the panko breadcrumbs in another shallow bowl. Whisk the egg with a splash of water in a medium bowl.
Use one hand to coat a piece of cheese in flour, then dip it in the egg. Let the excess egg drip off, then place in the breadcrumbs and use your other hand to coat the cheese with breadcrumbs. Set aside and continue with the rest of the cheese.
Heat a medium-large skillet over medium heat. Add enough avocado oil to coat the pan in a thin layer of oil, approximately ¼ cup.
When the oil is hot, use tongs to place the dredged grilling cheese into the skillet. Fry until golden brown and crisp, about 3 minutes.
Flip and continue frying the other side until golden brown and crisp, about 3 minutes. If desired, rotate the cheese pieces again and fry remaining sides of the cheese. Remove the fried cheese and place on a paper towel-lined plate.
Meanwhile, heat a small saucepan over medium heat. Add the honey, red pepper flakes, and cayenne. Bring to a low boil and simmer for about 5-10 minutes to let the honey infuse with the spices. Remove from heat.
Serve fried grilling cheese warm with hot honey as a dipping sauce.