We present a novel method for estimating the cosmic expansion rate $H_z$ at high redshift ($z approx 2$--$10$) using the internal dynamics and morphology of nearby galaxies—without relying on direct redshift—distance measurements. The approach models rotation curves of systems showing a clear structural transition between a central bar (interpreted as a fossil spiral) and an outer spiral disk. Within a dual-Lagrangian framework, these nested structures are assigned separate dynamical regimes, enabling $H_z$ to be extracted from the mass $M$ and emph{critical radius} $r_c$ of the proto-bar region. The critical radius is defined locally by the condition $v_H(r_c) = v_{m esc}(r_c)$, which yields a emph{local critical density} $ho_c propto H_z^2$. When this purely local relation is mapped onto cosmic time using the $Lambda$CDM matter-era scaling $H(t) propto t^{-1}$, it reproduces the universal $ho_c propto t^{-2}$ law. Applying the method to seventeen galaxies, and scanning early bulge mass fractions from $0.2%$ to $20%$ of today’s $M_{m bulge}$, we find that the $0.5%$—$5%$ range best matches the $Lambda$CDM timeline for disk—spiral onset. A refined backward-time minimization, incorporating a power-law bulge growth model, further narrows the plausible onset window to $0.4$—$1.8$~Gyr after the Big Bang. This technique complements high-redshift probes such as emph{JWST} imaging and CMB extrapolations, offering a new class of local, dynamical constraints on the early universe. If validated and applied to large rotation-curve samples, it could yield hundreds to thousands of independent $H_z$ determinations, refining both the cosmic expansion history and the baryonic structure formation timeline.