We discuss symplectic cutting for Hamiltonian actions of non-Abelian compact groups. By using a degeneration based on the Vinberg monoid we give, in good cases, a global quotient description of a surgery construction introduced by Woodward and Meinrenken, and show it can be interpreted in algebro-geometric terms. A key ingredient is the 'universal cut' of the cotangent bundle of the group itself, which is identified with a moduli space of framed bundles on chains of projective lines recently introduced by the authors.