What is the egg mixture used in gratins called in English?

This is one of these weird cases of synchronicity. I was wondering today how it is called in English. Then I answered a question, and needed the word for the answer.

The dishes I mean may not always be called gratins in English, sometimes they are called stew. They are a mix of diced vegetables and sometimes meat, mixed in a pan and put in the oven. When they are cooked enough (or when using parboiled vegetables), a mixture is poured over them. The mixture consists of eggs, a very small amount of flour, and one or more milk and fermented milk products. Soft or grated cheeses are sometimes mixed in or strewn on the surface to brown. The vegetables aren't swimming in the mixture, some of it seeps between them to fill the air pockets, but it mostly stays on top and sets and browns nicely.

I have only encountered this kind of dish in Mediterranean recipes. Typical examples are moussaka, ratatouile, gyuvech (although I know variants of all of them which don't have the mixture). In Germany, they'll be called a type of Auflauf, but an Auflauf is not necessary made with this eggy type of liquid, and I haven't heard a German name for the liquid either.

I wouldn't call it custard, as it is mainly used in savory dishes (although there are sweet variants where something else than vegetables is baked). Also, I haven't heard of custards without milk or cream, and these often contain something else, e.g. a mixture of yogurt and mashed feta. It isn't a batter either, the flour isn't enough to thicken it noticeably, and can sometimes be left out altogether. It is not a sauce, it is expected to set after baking.

Of course there is a probability that there isn't a name for it in English, because it isn't used often in English speaking countries. In this case, probably everybody could make up a well-sounding combination to refer to it, like "the egg mixture for moussaka crust". But I am not interested in such descriptive creations. What I am asking is, is there a single term which, when I say it without further qualifications, as in the sentence "I am making a [term]" will let my communication partner understand that I am cooking a dish of this kind and currently preparing the mixture for it.

Edit I don't mean a word which describes the content of the thing (although such a word would be nice too, if it were used only for this type of liquid and not for other liquids with similar content like custard). I am expecting something more of a purpose-based word. A comparable word would be "topping": A topping could be smoked ham or caramel sauce, depending on the dish. But it always has the same function. In this regard, I'd say that bechamel can be used as [term] sometimes, as in mfg's answer. But I wouldn't think of bechamel when I say the word. They are definitely not synonyms. "Custard" is problematic for the same reason: it has many other uses.