  We develop an Effective Field Theory (EFT) formalism to solve for the
conservative dynamics of binary systems in gravity via Post-Minkowskian (PM)
scattering data. Our framework combines a systematic EFT approach to compute
the deflection angle in the PM expansion, together with the 'Boundary-to-Bound'
(B2B) dictionary introduced in [1910.03008, 1911.09130]. Due to the nature of
scattering processes, a remarkable reduction of complexity occurs both in the
number of Feynman diagrams and type of integrals, compared to a direct EFT
computation of the potential in a PM scheme. We provide two illustrative
examples. Firstly, we compute all the conservative gravitational observables
for bound orbits to 2PM, which follow from only one topology beyond leading
order. The results agree with those in [1910.03008, 1911.09130], obtained
through the 'impetus formula' applied to the classical limit of the one loop
amplitude in Cheung et al. [1808.02489]. For the sake of comparison we
reconstruct the conservative Hamiltonian to 2PM order, which is equivalent to
the one derived in [1808.02489] from a matching calculation. Secondly, we
compute the scattering angle due to tidal effects from the electric- and
magnetic-type Love numbers at leading PM order. Using the B2B dictionary we
then obtain the tidal contribution to the periastron advance. We also construct
a Hamiltonian including tidal effects at leading PM order. Although relying on
(relativistic) Feynman diagrams, the EFT formalism developed here does not
involve taking the classical limit of a quantum amplitude, neither integrals
with internal massive fields, nor additional matching calculations, nor
spurious ('super-classical') infrared singularities. By construction, the EFT
approach can be automatized to all PM orders.
