In this article, we present a solution to the problem: "Which type of linear operators can be realized by the Dirichlet-to-Neumann operator associated with the operator $-\Delta-a(z)\frac{\partial^{2}}{\partial z^2}$ on an extension problem?", which was raised in the pioneering work [Comm. Par.Diff. Equ. 32 (2007)] by Caffarelli and Silvestre. In fact, we even go a step further by replacing the negative Laplace operator $-\Delta$ on $\mathbb{R}^{d}$ by an $m$-accretive operator $A$ on a general Banach space $X$ and the Dirichlet-to-Neumann operator by the Dirichlet-to-Wentzell operator. We establish uniqueness of solutions to the extension problem in this general framework, which seems to be new in the literature and independent interest. The aim of this paper is to provide a new Phillips-Bochner type functional calculus which uses probabilistic tools from excursion theory. With our method, we are able to characterize all linear operators $\psi(A)$ among the class $CBF$ of complete Bernstein functions $\psi$, resulting in a new characterization of the famous Phillips' subordination theorem within this class $CBF$.