"am cooking" is a valid translation. Koreans often use simple present to communicate an action currently taking place (what would be progressive or continuous tense in English). "Hamburger" could be singular without "a".-Steak is too expensive, so I cook hamburger. It could also be translated with an "a" or in the plural. Without context it is impossible to tell, so all should be acceptable.
(edit) i think another translation is "And so I cook hamburger.
(otherwise, In my neck of the woods, it is common to hear people say:
"Do you want hamburgers tonight? I'll cook some spicy ones. Bob, do you want me to cook a hamburger for you, too? How many do you want without salt? I have some frozen turkey patties, too!")
https://dict.naver.com/ would be a handy bookmark for studying the Korean language. You will find the following translations for "그래서"-- (and) so, therefore, accordingly, consequently; thus, for that[this] reason, on that account. 그래서 어떻단 말이야? So what?
However, you should also be aware that language is about communication. In real life, "so" is quite often used to start a sentence. I would guess the majority of English speakers are probably guilty of contravening the formal "rules" of the English language.