This is in fact the original Viginère method with a one-letter key.
The system as proposed by him, use a keyword $K$ of length $n$ for the first $n$ letters of a text, and then used the plaintext (from the start onwards) as the key for the rest. (it'a a "plain-autokey" cipher). He did not repeat the keyword, which is a somewhat weaker system, which is how his cipher is often represented nowadays.
It's still statistically weak, by autocorrelation it's easy to find the length of the keyword, and then it's reading in depth, essentially. Friedman wrote about this method on it in one of his cryptanalysis books.
Your friend's system only has a 1-letter (or one-alphabet) keyword, essentially. It doesn't matter if we use some random alphabet for the first, you just try all first plain letters and decrypt from the second letter onwards. Only one will make sense, normally. At least Viginère' system is not directly brute forcable in that way for long keywords.