In a food processor fitted with a metal blade, pulse the salmon fillet pieces 8 to 10 times until coarsely chopped but not pasty; transfer to a bowl. Add the remaining patty ingredients to the bowl, including ¼ teaspoon pepper, and stir gently to combine. Form six patties of equal size, each about ¾ inch thick. Cover and refrigerate the patties for 1 hour.
Prepare the grill for direct cooking over medium heat (350° to 450°F) and preheat a grill-proof griddle for 10 minutes.
Mix the cream cheese and the scallions until well blended. Refrigerate until ready to grill the patties.
Brush the patties on both sides with the oil. Place the patties on the griddle and cook over direct medium heat, with the lid closed, until fully cooked but still juicy, 6 to 8 minutes, carefully turning once. During the last 30 seconds to 1 minute of grilling time, toast the English muffins on the cooking grates, cut side down, over direct heat.
In a skillet over medium-high heat, melt the butter. Fry the eggs until the whites are set to your desired doneness.
Spread the scallion cream cheese on both sides of the toasted English muffins. Place a patty on each muffin bottom and top with an egg and a small handful of microgreens. Season with pepper. Serve warm with the muffin tops on the side.