We propose a matrix pencil based approach for design of output-feedback stabilizing controllers for a general class of uncertain nonlinear strict-feedback-like systems. While the dynamic controller structure is based on the dual dynamic high-gain scaling based approach, we cast the design procedure within a general matrix pencil based structure unlike prior results that utilized conservative algebraic bounds of terms arising in Lyapunov inequalities. The proposed approach models the detailed system structure and state dependence structure of uncertain terms. The design freedoms in the dynamic high-gain scaling based controller are extracted in terms of generalized eigenvalues associated with matrix pencils formulated to capture the detailed structures of bounds in the Lyapunov analysis. The proposed matrix pencil approach enables efficient computation of non-conservative bounds with reduced algebraic complexity and significantly enhances feasibility of application of the dual dynamic high-gain scaling based control designs.