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This paper introduces a scalable computational framework for optimal design under high-dimensional uncertainty, with application to thermal insulation components. The thermal and mechanical behaviors are described by continuum multi-phase models of porous materials governed by partial differential equations (PDEs), and the design parameter, material porosity, is an uncertain and spatially correlated field. After finite element discretization, these factors lead to a high-dimensional PDE-constrained optimization problem. The framework employs a risk-averse formulation that accounts for both the mean and variance of the design objectives. It incorporates two regularization techniques, the L0-norm and phase field functionals, implemented using continuation numerical schemes to promote spatial sparsity in the design parameters. To ensure efficiency, the framework utilizes a second-order Taylor approximation for the mean and variance and exploits the low-rank structure of the preconditioned Hessian of the design objective. This results in computational costs that are determined by the rank of preconditioned Hessian, remaining independent of the number of uncertain parameters. The accuracy, scalability with respect to the parameter dimension, and sparsity-promoting abilities of the framework are assessed through numerical examples involving various building insulation components.