Let \(P\) be a polynomial of degree \(d\geqslant 2\) and denote by \(K(P)\) the filled-in Julia set of \(P\), which is the set of points that have bounded orbit under iteration by \(P\). It is well-known that if all finite critical points of \(P\) escape to infinity, then \(K(P)\) is totally disconnected, while \(K(P)\) is connected if and only if all finite critical points have a bounded orbit. This paper focuses on the remaining intermediate cases, where only some of the critical points escape to infinity, and hence \(K(P)\) is disconnected but not totally disconnected. In this situation, if we pick an equipotential curve surrounding \(K(P)\) such that it passes through an escaping critical value, there is a preimage \(\gamma\) of this curve that contains the critical point and has a figure-eight shape. The restriction of \(P\) to each bounded complementary component \(U\) of \(\gamma\) defines what is known as a polynomial-like map \((f,U,U')\) of degree \(d'