Using Pollard's lambda in a key-recovery attack on DH
it simply tells a user to "calculate $y^*$", a calculation which itself appears to require several unknown values


Nope, it has $y^* = y / g^V$, where at this point, you know $y$, $g$ and $V$.

The original discrete log problem was given as $y = g^x$, and so you were given $y$ and $g$

In addition, $V = x \bmod z$, where $z$ is a smooth factor of $n$ (the group order, which the algorithm assumes you know). You can find the value of $V$ by taking the group order $n$, doing a partial factorization to obtain $z$ (even if it is infeasible to complete factor $n$, it's easy to find the small factors of $n$), and then by solving the Discrete Log problem $y' = g'^V$, where $g' = g^{n/z}$ and $y' = y^{n/z}$.  This particular Discrete Log problem can easily be solved by the Pohlig-Hellman algorithm, as $g'$ is in a subgroup of order $z$, which is smooth.