  We present the confirmation of two new planets transiting the nearby mid-M
dwarf LTT 3780 (TIC 36724087, TOI-732, $V=13.07$, $K_s=8.204$, $R_s$=0.374
R$_{\odot}$, $M_s$=0.401 M$_{\odot}$, d=22 pc). The two planet candidates are
identified in a single TESS sector and are validated with reconnaissance
spectroscopy, ground-based photometric follow-up, and high-resolution imaging.
With measured orbital periods of $P_b=0.77$ days, $P_c=12.25$ days and sizes
$r_{p,b}=1.33\pm 0.07$ R$_{\oplus}$, $r_{p,c}=2.30\pm 0.16$ R$_{\oplus}$, the
two planets span the radius valley in period-radius space around low mass stars
thus making the system a laboratory to test competing theories of the emergence
of the radius valley in that stellar mass regime. By combining 63 precise
radial-velocity measurements from HARPS and HARPS-N, we measure planet masses
of $m_{p,b}=2.62^{+0.48}_{-0.46}$ M$_{\oplus}$ and $m_{p,c}=8.6^{+1.6}_{-1.3}$
M$_{\oplus}$, which indicates that LTT 3780b has a bulk composition consistent
with being Earth-like, while LTT 3780c likely hosts an extended H/He envelope.
We show that the recovered planetary masses are consistent with predictions
from both photoevaporation and from core-powered mass loss models. The
brightness and small size of LTT 3780, along with the measured planetary
parameters, render LTT 3780b and c as accessible targets for atmospheric
characterization of planets within the same planetary system and spanning the
radius valley.
