We propose a reconstruction of the full $(x,\xi ,t)$ dependence of unpolarized isovector proton generalized parton distributions (GPDs) $H^{u-d}$ and $E^{u-d}$ from lattice QCD data in the pseudo-distribution formalism. For the first time, we extract double distributions (DDs) directly from lattice data, enforcing therefore an important property of GPDs linked to Lorentz symmetry. We use the flexible framework of multidimensional Gaussian process regression to regularize the inverse problem and present an assessment of the impact of model dependence on the systematic uncertainty. Our lattice ensemble corresponds to a pion mass $m_{\pi }=358$~MeV and a lattice spacing $a=0.094$~fm. We use larger hadron momenta, up to 2.7~GeV, and kinematic coverage compared to our previous computations and extract additional skewness-dependent moments of the GPD.