We aimed to create high-moisture plant-based meat alternatives (HPMAs) that closely mimic the fibrous structure found in conventional meat. To achieve this goal, we focused on the impact of modulating the heating and cooling die temperatures during the extrusion process on the structural properties of HPMAs. The optimal temperatures for the heating and cooling dies were determined to be 160 and 50 °C, respectively, following a comprehensive analysis of various quality characteristics. These included assessments of cutting strength, texturization degree, moisture content, and microstructure. Analyzing the correlation coefficients between heating and cooling die temperatures and the quality characteristics of HPMAs revealed that modulating the cooling die temperature has a greater impact on quality characteristics than modulating the heating die temperature. The quality characteristics of HPMAs were also compared with those of boiled and sous-vide cooked chicken breasts. The optimal HPMA exhibited the most similar fibrous macro- and micro-structure to sous-vide cooked chicken breast. Consequently, our study highlights the importance of modulating the heating die and cooling die temperatures, among various process parameters of high-moisture extrusion cooking, which is crucial for producing HPMAs closely resembling conventional meat.