Vtyjkyuk

Let $G$ be the free group with two generators, say $a$ and $b$.
Then the Cayley of $G$ with respect with the generating set is 4 regular tree.
If we contract every edge with the label $b$, then we end up with a tree $T$ such that $G$ acts on the edges of $T$ transitively.

Applying Bass-Serre theory implies that $G$ is the free product $langle arangle $ and $langle brangle $.

My question is the following:

How can I modify the Cayley graph of $G$ in order to get an HNN-exentension?

Actually, the tree that you have described is already an HNN extension tree, because its quotient graph of groups is a graph with one vertex and one edge. The vertex is labelled with the cyclic group $langle b rangle$, and the edge is labelled with the trivial group $E$.

The group element $a$ is the stable letter of that quotient edge, leading to the HNN presentation
$$F_2 = langle a,b mid aea^-1=e rangle
$$
where $e$ is the identity element of $langle b rangle$.

Although there are other seemingly less trivial HNN extensions of $F_2$ that can be described, and although this may seem like a somewhat trivial example of an HNN extension, it is nonetheless an example.