Melt butter in 2 quart (non-stick) sauce pan or larger.
Shred dried beef by hand into small pieces.
Add dried beef to melted butter and cook over medium-high heat until edges curl (about 10 minutes but watch carefully-don't burn it).
Stir in flour and mix thoroughly.
Add milk, reduce heat to medium low and continue stirring frequently until mixture is smooth & bubbly. (Depending on how thick you like it, it may take from 20 minutes to 1 hours to achieve desired consistency).
When it's as thick as you'd like, bring to a boil, stirring constantly and boil 1 minute.
Remove from heat and serve over toast and/or home fried potatoes.