Place the rack in the middle of the oven, then preheat the oven to 375°F.
Melt the butter in a small skillet over medium high heat. Saute the onion and the green pepper for roughly 4 minutes, stirring occasionally. Add the garlic and saute until fragrant, stirring constantly for 1 minute.
Brush olive oil on a 9x13-inch baking pan with sides. Whisk the eggs and milk together in a large bowl, then stir in the onion and green pepper mixture. Pour into the prepared pan and sprinkle the green onion, salt, and pepper on top. Bake for 22 to 25 minutes or until the center is set and the edges are slightly browned and a knife comes out clean. Let cool, then slice into 12 pieces.
While the eggs are baking, mix together the Dijon mustard, sour cream, and mayo in a small bowl.
Slice the English muffins in half and place on a cookie sheet cut side up. Toast the English muffins under the broiler until golden brown.
To assemble the sandwiches, start by putting one layer each of ham, egg, and cheese on top of a toasted English muffin. If you're serving the sandwiches right away, put the sandwich (without the top on) under the broiler until the cheese has melted to your liking.
Spread the mayo/Djion spread on the top half of the toasted English muffin and place on top of the sandwich. Serve warm.