  We compute perturbative QCD corrections to $B \to D$ form factors at leading
power in $\Lambda/m_b$, at large hadronic recoil, from the light-cone sum rules
(LCSR) with $B$-meson distribution amplitudes in HQET. QCD factorization for
the vacuum-to-$B$-meson correlation function with an interpolating current for
the $D$-meson is demonstrated explicitly at one loop with the power counting
scheme $m_c \sim {\cal O} \left (\sqrt{\Lambda \, m_b} \right ) $. The jet
functions encoding information of the hard-collinear dynamics in the
above-mentioned correlation function are complicated by the appearance of an
additional hard-collinear scale $m_c$, compared to the counterparts entering
the factorization formula of the vacuum-to-$B$-meson correction function for
the construction of $B \to \pi$ from factors. Inspecting the
next-to-leading-logarithmic sum rules for the form factors of $B \to D \ell
\nu$ indicates that perturbative corrections to the hard-collinear functions
are more profound than that for the hard functions, with the default theory
inputs, in the physical kinematic region. We further compute the subleading
power correction induced by the three-particle quark-gluon distribution
amplitudes of the $B$-meson at tree level employing the background gluon field
approach. The LCSR predictions for the semileptonic $B \to D \ell \nu$ form
factors are then extrapolated to the entire kinematic region with the
$z$-series parametrization. Phenomenological implications of our determinations
for the form factors $f_{BD}^{+, 0}(q^2)$ are explored by investigating the
(differential) branching fractions and the $R(D)$ ratio of $B \to D \ell \nu$
and by determining the CKM matrix element $|V_{cb}|$ from the total decay rate
of $B \to D \mu \nu_{\mu}$.
