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This paper develops state-feedback and output-feedback control design methodologies for boundary stabilization of a class of systems that involves cascade connection of a nonlinear viscous Hamilton-Jacobi partial differential equation (PDE) and a possibly unstable linear ordinary differential equation (ODE). First, an explicit state-feedback controller is designed based on the infinite-dimensional backstepping method and by utilizing a locally invertible feedback linearizing transformation whose role is to convert the non-linear viscous Hamilton-Jacobi PDE to a linear heat equation. Next, an output-feedback controller is proposed which makes the closed-loop system exponentially stable through the measured output of the system. The main feature of the proposed output-feedback scheme is that the distributed integral terms are avoided in the feedback law. This is achieved by exploiting the derived state-feedback controller and introducing a virtual ODE-PDE system, whose ODE state determines the stabilizing feedback law. Then, an observer is proposed that generates state estimates of the virtual system. In both schemes, local exponential stability is shown via Lyapunov analysis and an estimate of the region of attraction is provided. Finally, simulation examples for an unstable ODE are presented to validate the effectiveness of the proposed results.This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.