Ultra-intense X-ray sources have opened new avenues by creating new states of matter or probing and imaging living or inert matter. Free-electron lasers have a strong leadership by delivering pulses combining femtosecond duration and 10s of microJoules to milliJoule energy. However, these sources remain highly expensive limiting their number to a few worldwide. In parallel, laser-pumped soft X-ray lasers hold outstanding promises having demonstrated the most energetic monochromatic soft x-ray pulse and being intrinsically fully synchronized with any secondary source of the pump laser. Since the first successful demonstration of amplification of a high harmonic pulse in a plasma from gas in 2003 and from solid in 2008, we have developed an extensive numerical study. 2D hydrodynamic simulations showed that optimized Transient Collisional Excitation plasma amplifiers, may store up to 0.4 mJ in the population inversion. If carefully seeded, pulses of 80 fs and 20 mu J might be generated with table-top lasers (10J). As the energy extracted is far from the milliJoule requirements of most exciting applications, we studied the seminal experiment of Ditmire et al who seeded a plasma emitting milliJoules in the form of Amplified Spontaneous Emission (ASE).We retrieved and explained for the first time the experimental result (ASE 1,000 times stronger than amplified seed). We thus proposed and fully modeled the transposition of the so-called Chirped Pulse Amplification (CPA) in the soft X-ray range, showing that 6 mJ, 200 fs, fully coherent soft X-ray pulse is accessible with compact pump lasers.