Probing the \textit{Kepler} hottest small planets via homogeneous search and analysis of optical secondary eclipses and phase variations.

We perform a homogeneous search for and analysis of optical occultations and phase variations of the most favorable ultra-short-period (USP) ($P<1$~d) sub-Neptunes ($R_{p}<4 R_{\oplus}$) observed by $\textit{Kepler}$ and K2, with the aim of better understanding their nature. We first selected 16 $\textit{Kepler}$ and K2 USP sub-Neptunes, based on the expected occultation signal. We filtered out stellar variability in the $\textit{Kepler}$ light curves, using a sliding linear fitting and, when required, a more sophisticated approach based on Gaussian Process regression. We simultaneously modeled the primary transit, secondary eclipse, and phase variations in a Bayesian framework, by using information from previous studies and knowledge of the Gaia parallaxes. We confirm the optical secondary eclipses for Kepler-10b ($13\sigma$), Kepler-78b ($9.5\sigma$), and K2-141b ($6.9\sigma$), with marginal evidence for K2-312b ($2.2\sigma$). We report new detections for K2-106b ($3.3\sigma$), K2-131b (3.2$\sigma$), Kepler-407b ($3.0\sigma$), and hints for K2-229b (2.5$\sigma$). For all targets with the exception of K2-229b and K2-312b, we also find phase curve variations with confidence level higher than $2\sigma$. Two USP planets, namely Kepler-10b and Kepler-78b, show non-negligible night-side emission. This questions the scenario of magma-ocean worlds with inefficient heat redistribution to the night-side for both planets. Due to the youth of the Kepler-78 system and the small planetary orbital separation, the planet may still retain a collisional secondary atmosphere capable of conducting heat from the day to the night side. Instead, the presence of an outgassing magma ocean on the day-side and the low high-energy irradiation of the old host star may have enabled Kepler-10b to build up and retain a recently-formed collisional secondary atmosphere.