One of the primary targets for the new lignocellulosic feedstock-based biorefinery is the simultaneous valorization of holocellulose and lignin. Acidified organosolv treatment is among the most promising strategy for recovering technical lignin, water-soluble hemicellulose, and cellulose pulp with increased accessibility to hydrolytic enzymes. In this work, a design-of-experiment (DoE) approach was used to increase the cellulose recovery, digestibility, and the delignification of Cynara cardunculus L. feedstock. In the first treatment, the milled biomass was subjected to microwave-assisted extraction using an acidified GVL/water mixture to separate lignin and hemicellulose from cellulose. In the second treatment, the cellulose pulp was hydrolyzed by cellulolytic enzymes to demonstrate the enhanced digestibility. At the optimal condition (154 °C, 2.24% H2SO4, and 0.62 GVL/water ratio), the cellulose pulp showed a cellulose content of 87.59%, while the lignin content was lower than 8%. The cellulose recovery and digestibility were equal to 79.46% and 86.94%, respectively. About 40% of the initial hemicellulose was recovered as monosaccharides. This study demonstrated the effectiveness of the two-step organosolv treatment for biomass fractionation; however, as suggested by DoE analysis, a confirmative study at a low temperature (<154 °C) should be performed to further increase the cellulose recovery.