  The recent measurements of baryon acoustic oscillations (BAO) from the DESI
collaboration have presented an indication for dynamical dark energy, when
adopting the $(w_0,w_a)$ parametrization of the equation of state. The
associated posterior constraints imply a crossing of the phantom divide. The
latter, however, has profound theoretical implications because not all models
can do so without developing incurable instabilities. Simple quintessence
models of dark energy, for instance, would be ruled out if such a crossing is
confirmed. We perform a non-parametric reconstruction of the equation of state,
and confirm that crossing of the phantom divide is required by the DESI BAO
data. We then explore the theory space of Horndeski gravity employing a
reconstruction method based on the effective field theory of dark energy, and
show that for most of the models it is still difficult to safely cross the
divide. We identify non-minimal coupling to gravity as the key modification
which sustains a stable phantom crossing in the general Horndeski theory space
and fits DESI observations. Guided by these insights, we propose the
\textit{Thawing Gravity} model which has the same number of parameters as
$w_0w_a$CDM and naturally realizes non-minimal coupling when dark energy
becomes non-negligible. \textit{Thawing Gravity} improves the fit over
$\Lambda$CDM for DESI BAO, CMB as well as type Ia Supernovae.
