It is well-known that reliable generation of high-quality meshes that fit the boundaries of complex domains can be challenging, especially when the mesh comprises quadrilateral or hexahedral elements. To address this difficulty, a set of Nitsche-based methods that utilize unfitted meshes has been proposed, thereby avoiding complex meshing. However, unfitted mesh methods inevitably need to address the complications associated with enforcing boundary conditions on arbitrary boundary surfaces. Unlike Nitsche-based methods such as CutFEM and aggregated FEM, the cut-cell finite element method propounded by Papadopoulos et al. is the first unfitted mesh method that strongly enforces Dirichlet boundary conditions. This method has been demonstrated to be robust with linear elements in both two-dimensional and three-dimensional domains and is easy to implement. Due to these attractive features, a higher-order generalization of this method is presented in this work. Specifically, the proposed method strongly and primally enforces the Dirichlet boundary conditions via a projection strategy, indicating that the prescribed values are incorporated by directly constraining the corresponding degrees of freedom in the algebraic system, rather than introducing additional equations or auxiliary variables. To ensure well-conditioning, a set of dimensionless parameters is introduced to control the projections, with parameter values predetermined using a genetic algorithm and held fixed throughout all numerical tests. Furthermore, the Neumann boundary conditions are enforced weakly using an isoparametric higher-order triangulation without introducing any additional degrees of freedom. Through extensive numerical experimentation, this approach is demonstrated to be robust and to achieve the optimal rate of convergence across a variety of representative boundary-value problems. In addition, adaptive strategies within this framework are currently under development to further improve accuracy for problems with localized complexity.