Start with preparing a roux. To do this, melt the butter in a thick-bottomed saucepan (or Dutch oven) and add the flour. Stir well for a minute until it starts to turn just a little golden.
Gradually start stirring in the beef broth. It's best to incorporate it 1/2 cup at a time to avoid any flour clumps from forming.
When the roux starts to bubble, turn the heat down to the lowest setting, cover the pan with a lid and simmer the sauce for 20 minutes or so until thickened.
Take the sauce off the heat and add in the sugar and heavy cream until homogenous then stir in the vinegar and chopped dill as well.
Taste the sauce and adjust with more salt or sugar according to your preference.
Let the sauce sit for 5 minutes to steep in all the flavors.
Serve warm, drizzled over dumpling slices and hard boiled eggs or meat!