Advanced manufacturing processes subject materials to complex thermal histories, characterized by extreme heating and rapid cooling rates. These processes significantly influence the final microstructure. In order to address these challenges, the present work proposes a comprehensive multi-physics material model derived from the extended Hamilton principle. This principle establishes a unified variational framework for coupled thermal and mechanical processes with phase transformations at finite strains. The capabilities of the framework are demonstrated using the additive manufacturing of glass as a representative application. In this application, the material integrity depends on the intricate interplay of thermodynamic driving forces and kinetic limitations. The formulation integrates a rigorous thermodynamic description of first-order melting and second-order glass transitions. The methodology employed in this study involves a kinematic split, a technique demonstrated to effectively account for thermal expansion, viscous flow, and phase-specific density changes. A temperature-dependent viscosity model is employed to capture the kinetic freezing of the microstructure during vitrification and viscoelastic flow. The numerical implementation utilizes a monolithic Neighbored Element Method (NEM) to solve the heat equation, thereby ensuring computational efficiency and stability. The mechanical constitutive laws are implemented in ANSYS via user subroutines. Numerical investigations at the material point level validate the model by reproducing Time-Temperature-Transformation behavior under varying cooling rates. In addition, three-dimensional finite element simulations have been utilized to demonstrate the accumulation of residual stresses and macroscopic warpage resulting from the interplay between phase transformation kinetics and viscous relaxation.