  We present the first application of QCD light-cone sum rules (LCSRs) with
$B_{(s)}$-meson distribution amplitudes to the $B_{(s)}\!\to\! D_{(s)0}^*$ form
factors, where $D_{(s)0}^*$ is a charmed scalar meson. We consider two
scenarios for the $D_0^*$ spectrum. In the first one, we follow the Particle
Data Group and consider a single broad resonance $D_0^*(2300)$. In the second
one, we assume the existence of two scalar resonances, $D_0^*(2105)$ and
$D_0^*(2451)$, as follows from a recent theoretically motivated analysis of
$B\to D\pi\pi$ decays. The $B\!\to\! D_0^*$ form factors are calculated in both
scenarios, also taking into account the large total width of $D_0^*(2300)$.
Furthermore, we calculate the $B_s\!\to\! D_{s0}^*$ form factors, considering
in this case only the one-resonance scenario with $D_{s0}(2317)$. In this LCSRs
calculation, the $c$-quark mass is kept finite and the $s$-quark mass is taken
into account. We also include contributions of the two- and three-particle
distribution amplitudes up to twist-four. Our predictions for semileptonic
$B\!\to\! D_0^*\ell\nu_\ell$ and $B_s\!\to\! D_{s0}^*\ell\nu_\ell$ branching
ratios are compared with the available data and HQET-based predictions. As a
byproduct, we also obtain the $D_0^*$- and $D_{s0}^*$-meson decay constants and
predict the lepton flavour universality ratios $R(D_0^*)$ and $R(D_{s0}^*)$.
