The challenge of developing highly thermally conductive polymeric composites to meet the growing thermal management demands has recently attracted a lot of attention. To achieve a through-plane thermal conductivity higher than 2 W/mK, a high filler concentration within the poly(ether ether ketone) (PEEK) matrix is required, thus adding to the complexity of polymer processing. In this study, an optimized twin-screw extrusion melt-compounding technique was developed by tuning the melt flow of unfilled PEEK, feeding zones, and process cycles for dispersion of hexagonal boron nitride (h-BN) in the PEEK polymer. The prepared composites demonstrated exceptionally high in-plane and through-plane thermal conductivity of 12.451 and 2.337 W/mK, respectively, at 60 wt % h-BN loading. This improvement of thermal conduction in both directions can be attributed to two factors: (1) formation of through-thickness surface contacts between h-BN particles due to shear-driven exfoliation during compounding stage and (2) high degree of alignment of h-BN platelets achieved during molding stage. The calorimetric and thermogravimetric analyses indicated that the prepared composites possess enhanced crystallinity compared to unfilled PEEK and are thermally stable in elevated temperature ranges. The rheological characterization exhibited a progressive increase in viscosity and shear-thinning behavior of composite melts proportional to the h-BN concentration. Using the temperature and time-dependent rheological results, viscosity buildup profiles were constructed to outline allowable melt viscosity ranges for each composite composition. These profiles can be utilized to tailor the residence time of a composite melt by varying the filler concentration and process temperature during advanced manufacturing processes such as extrusion-based additive manufacturing and powder bed fusion. Hence, we provide a facile and industrially scalable method for development of h-BN-filled PEEK composites with high thermal dissipation characteristics aimed for thermal management in various harsh environment applications.