À la Lyonnaise is a French cooking term used to describes a dish that uses onions in some way.
In the grande cuisine sense, it means either a side accompaniment of onions that are stuffed and braised, plus potatoes, or garnished with chopped onions that have been sautéed in butter, then had a bit of vinegar and chopped parsley stirred into them. It’s the last two bits that would, sadly, disqualify the eggs and onion that you get on toast at your standard Jewish deli from being “oeufs À la Lyonnaise.”
Lyonnaise sauce, made with reduced white wine and a demi-glace, is called Lyonnaise because it has onions in it.
“À la Lyonnaise” means in the style of Lyons, France.