This is what’s typically called “boiled dressing.” Which is sort of a misnomer, because you don’t actually boil everything together, because if you did, you’d scramble the eggs.
Boiled dressing was especially popular before mayonnaise was commercially sold in the early 20th century. Miracle Whip, in fact, was intended to be a blend of the more expensive mayonnaise with the less-expensive boiled dressing.
Here’s a version from the June 16, 1870 edition of the Waukesha (Wisconsin) Freeman:
Salad dressing.–To make a boiled salad dressing for chopped cabbage, which is very appetizing and healthful. Six eggs well beaten; six tablespoonfuls of melted butter; six tablespoonfuls of sour or sweet cream; two tablespoonfuls of mustard; one of pepper, one of salt, and two coffee-cups of vinegar. Boil all but the eggs; when thoroughly mixed, stir them in quickly, until it thickens. Then take from the fire, and set the dish in cold water. Be sure that the eggs do not curdle. This makes an excellent dressing for celery, lettuce, chicken and lobster salad. Olive oil can be substituted fro the cream or butter. It is nice turned over cold chopped potatoes, mixed up with a little raw onions.
From the box of A.D. from Lutz, Florida, by way of Pennsylvania in the 1940s, and originating in Ohio in the 1920s.
1 tsp. mustard
1/4 tsp. salt
2 heaping tsp. sugar
1/4 tsp. pepper
1 tsp. butter
Beat all together and set in a double boiler. When it begins to thicken add 1/2 cup vinegar and stir until it is the consistency of cream.