Quantum gravity of a brane-like Universe is formulated, and its Einstein limit is approached. Regge-Teitelboim embedding of Arnowitt-Deser-Misner formalism, parametrized by the coordinates yA (t, xi), is governed by some ρAB(y, y′, y″). Invoking a novel Lagrange multiplier λ, accompanying the lapse function N and the shift vector Ni, we derive the quadratic Hamiltonian script H sign = 1/2 N [PA((ρ - λI)-1)AB PB + λ] + Niy,AiPA. The inclusion of matter resembles minimal coupling. Setting PA = -i(δ/δyA), we derive a bifurcated Wheeler-Dewitt-like equation. Einstein gravity, associated with λ being a certain fourfold degenerate eigenvalue of ρAB, is characterized by a vanishing center-of-mass momentum ∫ PA d3x = 0. Troublesome (ρ - λI)-1 is then replaced by regular M-1, such that M-1 (ρ - λI) defines a projection operator, modifying the Hamiltonian accordingly.