Bake the fingerling potatoes at 350°F for 20 minutes or until just tender. Remove from oven and chill at least 2 hours.
Place the corned beef in the bowl of a stand mixer fitted with a paddle attachment. On low speed, break up the corned beef until it resembles coarsely pulled pork. Work in batches if necessary.
Sauté the peppers and onions with the butter until softened. Do not color. Cut the chilled fingerling potatoes into 1" pieces.
In a large bowl, combine the corned beef with the vegetables and potatoes and mix well. Add just enough mashed potato until the mixture loosely holds it shape when compressed. Season with salt and pepper to taste. Scale portions to 8 oz.
Using a ring mold or hamburger patty form lined with film wrap, press the mixture until it is firm and holds the shape of a hamburger patty. Finish wrapping and chill for 12 hours, up to 5 days. Do not freeze.
Using a non-stick pan with a little vegetable oil or well-seasoned griddle, sear one side of the patty until dark golden brown and crispy. Using a full-size spatula, carefully turn the patty over without disturbing its shape and brown the other side. Continue cooking until an internal temperature reaches 165°F.
Carefully transfer the patty (it will be very delicate when hot) to serving plate and top with poached or fried eggs. Serve immediately.