Preheat the oven to 400°F.
Add potatoes to a large pot of cold water. Add salt to the water. Bring to a boil over high heat and cook until potatoes are tender, 10-20 minutes.
Remove pot from heat, drain the potatoes, then return potatoes to the pot. Add butter and milk to the pot. Mash potatoes until smooth. Add salt and freshly ground pepper, to taste. Cover and set aside.
Heat butter in a large skillet over medium heat. Add the onions and cook until softened and starting to turn golden. Add the Worcestershire, ketchup, chicken broth, parsley, and thyme to the skillet. Simmer over medium heat until the liquid reduces by half.
Add the cooked corned beef to the skillet and reduce heat slightly. Simmer for 3-4 minutes or until there is only a bit of liquid left in the mixture. Taste the mixture and add additional salt and pepper, as needed.
Spoon filling into a small baking dish. Spoon mashed potatoes over-top in an even layer.
Top with shredded cheese, then sprinkle with bread crumbs.
Bake in preheated oven for about 30 minutes, or until filling is heated through and bubbling and top is golden.
Let the pie cool for a few minutes before serving.