I used to sing raspy vocals for about 75% of my heavy songs/ hard rock songs but I’m not sure if I ever did the full on Cookie Monster thing. I found that singing with a rasp, especially if you’re singing with a lot of power, is very taxing and damaging on the throat. When I was gigging regularly I would avoid doing much raspy singing for live performances. I’d sing mostly with a clear voice. Singing hard rock/ classic rock, …anything loud and aggressive can quickly lead to damage of the vocal cords and tissues of the throat. I’ve always loved a raspy, aggressive vocal sound though. I’m not a big fan of the soft, wimpy, straight ahead vocal sound of guys like Bono. He sounds so casual and pedestrian as if he’s always taking it easy. It just sounds so ‘un-rock’n’roll”, as opposed to singers like Lou Gram and Paul Stanley who used to belt out powerful and often times raspy vocals. They had power, great vocal range and rasp. They could also sing crystal clear and gentle… Pavarotti and many of those tenor opera singers have that same kind of power, though I’m not sure if I’ve ever heard them using “rasp”. Paul Stanley and Lou Gram have/ HAD’ the same attributes as those opera singers. They’re able to maintain a lot of girth and fullness in their voices even when they’re stretching quite high into their upper registers.
My guess regarding the best method to achieve Cookie Monster vocals would be to use a dynamic mic in very close proximity to the singer’s mouth (a large diaphragm condenser would probably work just as well, though you might have to be careful not to overdrive the capsule too much with sound pressure because of the close proximity and volume. Maybe even singing with the mouth touching the windscreen/ grill of the mic with the singer not particularly focusing on power/ volume but mostly the guttural growl from their throat. I’ve seen a recording demonstration where one guy was probably 8 to 12 inches away (maybe further) and he did a very good job of doing the Cookie” thing. I should try it out some day just to see which way works the best. I probably wouldn’t have much use for that style of vocal but I like to experiment with different styles just for fun and learning.