Taking a quick break from research-reading to share the recipe dragonknitter asked for.

This is the Good Housekeeping Cook Book (copyright 1933) version of Tomato Rarebit. It's similar to a dish I make sometimes from "Square Meals" which the Sterns call "Rinktum Tiddy" in that cookbook. (In their book, it is in the chapter on Nursery Foods).

This recipe serves six; I would make 1/3 of it (the smallest amount one can reasonably make, as you will see)

3 T butter
3 T flour
1 1/2 c. milk
1 1/2 c. "canned tomato sauce or canned condensed tomato soup". (I am far more prone to use the canned "petite diced" tomatoes. Condensed soup doesn't sound so good for this - too salty. Even better, of course, would be homecanned tomatoes if you have them).
1/4 t baking soda (I never use this. It is in a lot of old tomato recipes, I suppose to cut the acid. Modern tomatoes are less acid than they were even 50 years ago [due to selective breeding for sweeter strains] and anyway, the acidity doesn't bother me)
3 eggs, slightly beaten
3/4 t salt (or less, especially if you're using canned tomato soup)
3/8 t dry mustard
"speck" cayenne (I'd probably go with 1/8 t)
3/4 lb grated cheese (3 cups)

You melt the butter in a double boiler (or a saucepan if you can be very careful and keep a low flame under it). Mix the flour with the butter until smooth. Then add the milk and stir until it is thickened (in other words: make a white sauce). Then add the tomatoes (and the soda if you are using it). Beat the eggs in a bowl and add the spices and salt. Then, carefully temper the eggs by putting a little of the warm tomato mixture into it and stirring - it is probably best to do this in a couple steps. (I didn't do that the first time I ever made the thing - totally forgot - and wound up with sort of boiled-scrambled eggs in tomato sauce). When that is all mixed, fold in the cheese and heat until the cheese is melted.

This is traditionally served over crackers or toasted bread; I generally prefer to put the rarebit in a bowl and make "toast fingers" separately to dip in the rarebit. You can also eat it over rice if you like.

I make about 1/3 of this recipe - one egg. It makes two quite generous servings. (I suppose in the 30s, more people walked everywhere and homes weren't as well-heated, so maybe people needed more calories).

You can do the white sauce with just cheese in it (and the cayenne and mustard) and you have sort of a Welsh rarebit, though most of the true cheese rarebit recipes I've seen also call for ale.