Adapted from The Big Green Egg® Cookbook
To make the avocado relish, using a fork, lightly mash the avocado in a small bowl. Add the tomatoes, dill, lemon juice, ¼ teaspoon salt, and 1/8 teaspoon pepper and toss. Set aside.
Cut the croissants in half lengthwise, brush with 2 tablespoons of the butter, and place, cut side down, on the Grid. Close the lid of the EGG and toast lightly. Transfer the croissants to a platter and set aside.
Brush the Redfish fillets with 2 tablespoons of the butter. Season each fillet with Old Bay seasoning, salt, and pepper. Place the Redfish on the exposed Grid. Melt the remaining 2 tablespoons butter on the Griddle and crack the eggs into the butter. Close the lid of the EGG and grill the Redfish and eggs for 3 minutes, turn both the eggs and Redfish over, and continue cooking for 2 more minutes. The interior of the fish should be opaque. Transfer to a rimmed sheet pan.
To assemble, place a piece of Redfish on the bottom half of a toasted croissant and top with an egg and a slice of cheese. Repeat this process for the remaining croissants. Return the croissants to the Grid, close the lid of the EGG, and let the cheese melt for 30 seconds. Spread 2 tablespoons of relish on top of each sandwich. Place the top of the croissant on the sandwich and serve immediately.