To make the horseradish cream, whisk the sour cream, horseradish, chives, salt, and pepper in a small bowl until blended. Set aside.
Melt 4 tablespoons of the utter in a small saucepan on the stovetop over low heat. Using a pastry brush, spread the muffin halves with butter. Place the muffin halves on the Griddle, cut side down, until toasted and lightly browned. Using a long-handled spatula, transfer the muffins to a platter. Spread each of 4 muffin halves with 2 teaspoons of the horseradish cream. Set aside.
Brush all the tomato slices first and then the steaks with butter, and season with salt and pepper. Place the steaks on the Grid and, while they are cooking, melt the remaining 2 tablespoons of butter on the Griddle. Crack the eggs onto the hot Griddle. Close the lid of the EGG and cook for 3 minutes, or until the whites of the eggs are set. Using a long-handled spatula, turn the steaks and eggs over and top each egg with a slice of cheese. Close the lid of the EGG and continue to cook for 2 minutes or until the cheese is melted. Using a long-handled spatula, remove each steak and place it on the bottom half of an English muffin. Top each steak with 1 egg, a slice of tomato, and the top of the English muffin. Place the assembled sandwiches on the Grid.
Close the lid of the EGG and heat for 1 minute, until the sandwiches are hot.
Transfer the sandwiches to a platter and serve immediately.
—From Big Green Egg Cookbook/Andrews McMeel Publishing
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