Grandma's Welsh Rarebit, Written Down at Last

When my Grandma migrated from England to the United States, she didn't bring many material possessions, but she did carry of wealth of stories, traditions and recipes in her head. Of her recipes, Welsh Rarebit was one of my favorites.
Exotic though it sounds, Welsh Rarebit is basically a cheese sauce, with, in Grandma's verison, the slightest hint of mustard. It can be served over toasted bread or English muffins or with a myriad of other possibilities.
The problem with Grandma's recipe was that it remained in her head. She wrote many of her other recipes down but not that one. When I was growing up and wanted to make it, usually on a summer afternoon for lunch, I'd run next door to Grandma's house.
"Grandma," I'd say, "how do you make Welsh Rarebit again?" She'd explain the procedure. "Wait," I'd say, "isn't there an egg in it?"
"You can add one if you'd like," she'd tell me.
That's the kind of recipe it was, a vague sort of this, this, and this. And possibly this. And that's the way I've always made it--until now.
Since I'm determined to pass a copy of my recipes onto posterity, I've written it down, some of its vagueness still intact.

Welsh Rarebit

Servings 5

Cook Time: 10 minutes

A recipe from Grandma Harris, to be served over brioche, toasted bread, English Muffins, or with eggs and ham, Eggs Benedict style. Also good as a pasta sauce.

1 1/2-2 tablespoons butter

1 1/2-2 tablespoons flour

1 1/2 cups milk

1/2 to 1 teaspoon mustard to taste

2 cups shredded cheese

1 egg, optional 1/2 teaspoon salt, or to taste Pepper, to taste

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In a saucepan, melt the butter, then add the flour. (One and a half tablespoon of each for a thinner sauce, two, for thicker). Cook and stir for one minute. Add the milk and stir with a whisk until the mixture begins to bubble and no lumps remain. The process can be sped up by heating the milk first in the microwave. Add the mustard and cheese. Cook and stir until the cheese melts. If using the egg (it enriches and thickens the sauce even more), crack it into a small bowl and beat it. Add a small amount of the hot sauce in with the egg to temper it, then pour the egg mixture back into the saucepan. Cook and stir over medium heat for two more minutes. Add salt and pepper, to taste.

It can be served like this: over a toasted English muffin.

Or you can stack a few things on top. Ham, for example.

 Followed by poached eggs. (Seen here in the tray from the egg poacher I got for Christmas. Love it!)

 And then the sauce on top. A cheesy version of Eggs Benedict. My family's favorite way.