Melt butter in a medium saucepan over medium heat; simmer, stirring frequently, until butter is browned, fragrant, and foamy, with brown bits in it, 5 to 10 minutes.

Add chicken broth and bouillon to the pan with the butter; bring to a boil over high heat. Stir in egg noodles, bring to a second boil, cover, and turn off heat; leave pot on the same burner. Let stand for 30 minutes, stirring 2 or 3 times.