Yes. You should do a google search on making butter -- or better yet on making cultured better. Then, make the butter.
Yes. Follow the recipe. After you've tried it, adjust to suit your preferences. Learning how to figure out how a recipe will work, and what variations you want to make before trying it is an important part of learning to cook. However, you don't seem to be at that level of expertise with Generation Madeleines yet. Better to stay close to the recipe -- the first time anyway.
Not if I understand what is meant by "fermented butter." The problem seems to be one more of language translation than of very exotic ingredients. First we have to figure out what the recipe originator meant, than supply the right ingredients. In this case, it appears (to me, anyway) that cultured butter is called for.
As a WORD (not as an ingredient or process), "cultured" became "fermented," and "fermented" became "yeast." In the case of "cultured" butter, the culturing agent is a lacto-bacterium (bacteria that affects milk) and not a yeast spoor. The bacteria and spoors are both eukaryotes (single-cell beasts), but are otherwise different. They won't change milk in the same ways. Furthermore, you don't want live yeast in a cake/cookie batter like a Madeleine. Without several rises and/or a great deal of working , the yeast would give the Madeleine's a very unpleasant taste; but if you did work the dough and/or allow for several fermentations you'd have entirely the wrong crumb. Even then, there are better ways to get fermented yeast into dough than by mixing it into butter, by way of a poolish or a sour starter to name two methods.
I think once you try cultured butter -- whether bought or DIY -- spread on a piece of toasted bread, the whole thing will become clear. It's got a nutty, tangy taste and is the kind of ingredient European and exotic enough to look very sexy to Asian bakers. It makes a lot of sense within the concept implied by the rest of the ingredient list and the name for Generation Madeleines.
It's possible I'm wrong and that "fermented butter" depends in some way on a yeast reaction. In case it's not obvious by now I'm giving an educated guess regarding the ingredients. I'm certainly not an expert on Generation Madeleines, and until this thread, never heard of "yeast butter." I've made regular Madeleines though, am familiar with the normal ingredients, have a good grasp of ingredients generally, and am familiar with the techniques involved. Take it for what it's worth -- a guess from a pretty good cook. Not the word of God handed down on stone tablets to the pastry class meeting Tu., and Th., 10am - 12am, Mt. Sinai kitchen.
If I'm wrong, it won't be the first time and I won't be embarassed,