In a small bowl add all the ingredients for the dressing and mix until combined. Refrigerate until ready to assemble the sandwich.
Get the instructions on how to make homemade hash browns here. You can absolutely use store-bought.
Preheat a non-stick pan to medium-high and add 1 tablespoon oil and 1 tablespoon unsalted butter. Add the diced onions and cook until starting to brown. About 5-8 minutes. Place in a bowl and set aside.
In a non-stick pan, melt 1 teaspoon butter on medium-low until bubbles start to form. Add the egg and cook until the egg whites start to set and the yolk is still runny. About 3-4 minutes. Sprinkle with salt, pepper, and fresh thyme (optional).
Slice the corned beef into strips and add it to a medium pan with 1 tablespoon of water. Put on the lid and heat through about 3 minutes.
Time to assemble, add the Russian dressing to the top and bottom bun. Then, add the sauteed onions, corned beef, hash brown, a slice of cheese, and egg. Continue until all sandwiches are made. Time to enjoy!