Can I make a hot mayonnaise?
You are basically talking about a Hollandaise sauce, which is similar to mayonnaise in that it is an emulsion of egg yolk and fat. The main difference is that in mayonnaise the fat is oil, whereas in Hollandaise it is butter. The main flavours in mayonnaise come from an acid (lemon juice or vinegar) and mustard - Hollandaise likewise uses lemon juice, and mustard can be easily added.

Recipes abound for Hollandaise, so you will have no problem finding one on Google.