The microscopic mechanism of high-temperature cuprate superconductors remains a central enigma in condensed matter physics. Deciphering the superconducting pairing mechanism—specifically, elucidating the intricate interplay between the rich tapestry of competing orders in a doped Mott insulator and the emergent superconducting state—is fundamental to solving this long-standing puzzle. However, progress has been severely hindered by the lack of suitable material platforms that allow for such detailed investigation. In this study, we establish the halide-based cuprate Ca₂CuO₂Cl₂ (CCOC) as a definitive platform to address these stringent requirements, leveraging high-pressure and high-temperature synthesis to grow a series of superior-quality single crystals. By systematically modulating the nominal dopant concentration within the precursor under invariant pressure conditions (5 GPa), we achieved meticulous control over the resultant carrier density, culminating in two pivotal advancements. First, we synthesized an extensive spectrum of Na⁺-doped (hole-type) CCOC crystals, establishing via rigorous SEM-EDX analysis a quantitative correlation between nominal and actual doping concentrations. This revealed a saturation behavior at higher nominal values, with a maximum attainable hole concentration of approximately 0.12 under our 5 GPa conditions. Magnetic susceptibility measurements demonstrated that these crystals span the critical region of the phase diagram, traversing from the parent Mott insulator through the underdoped regime (p < 0.07), with superconductivity emerging at p ≈ 0.08 (T_c ≈ 12 K) and monotonically increasing to T_c ≈ 19 K at the maximum achieved doping of p ≈ 0.12, thereby approaching the anticipated optimal doping level. Second, we achieved the inaugural synthesis of electron-doped CCOC single crystals via substitution of Ca²⁺ with trivalent lanthanides (Nd³⁺ and La³⁺). SEM-EDX analysis unequivocally confirmed successful incorporation, yielding actual electron doping concentrations of approximately 1.5%. Although these specific crystals do not exhibit superconductivity down to 2 K, they represent a groundbreaking proof-of-concept for electron doping within this material family. By constructing the superconducting phase diagram and directly correlating actual doping concentration with T_c, our single-crystal data exhibits excellent agreement with previously reported polycrystalline results, thereby validating our synthetic methodology. Moreover, the distinctive layered architecture of CCOC, characterized by weak interlayer coupling imparted by the apical chlorine atoms, renders these crystals exquisitely amenable to cleavage, consistently producing large, atomically pristine surfaces indispensable for incisive surface-sensitive spectroscopic investigations. This CCOC platform, uniquely integrating continuously tunable hole doping with the first-ever realization of electron doping in an identical host lattice—combined with its superior cleavage properties—provides a powerful and transformative system to systematically probe the evolution of electronic structure and competing orders with doping and carrier type, directly paving the way to decipher the enigmatic mechanism of high-temperature superconductivity.