How to catch up on years of missed prayers?

Assalamu alaikum wa rahmatullah. I learned that, according to most scholars, I have to make up for prayers I missed years ago. I started doing it. I pray my current fard and add a make-up fard after it. Then I keep going beyond that fard. Example: Dhuhr time comes. I pray the fard, then a make-up Dhuhr fard. Then I continue. I do a make-up Asr, Maghrib, Isha, and Fajr. So I knock out a whole missed day right after each current fard. Recently someone told me this might be wrong. They said I should just make up the missed prayer with the same prayer. Like, when Dhuhr is called, pray it and then only the missed Dhuhr. That’s it. Wait for Asr and do it again. I’m not sure about this way because I can’t find any sources. I have a lot of years to cover and I don’t think I’d finish if I only do one extra day a day. We don’t know when we’ll pass and I’m scared of not doing this duty as much as I can. Also, I recently found out that if your period starts after a prayer time begins and you had time to pray but didn’t, you have to make it up. I’ve never heard this and I don’t know which prayers this applies to. So my question: How should I make up my missed prayers? Do I include prayers missed during my period? If yes, how?