Organic two-dimensional (2D) materials hold great potential in a broad range of applications. However, their practical utility is constrained by limited structural diversity and a lack of synthetic strategies. Herein, we report that simply stirring immiscible solutions of poly(propylene glycol)bis(2-aminopropyl ether) and 1,3,5-benzenetricarbonyl trichloride results in a dynamic microinterfacial polymerization that affords novel 2D polymer sheets (designated as PEO-BTA). These sheets can be transformed to Na-ion conducting materials via treatment with sodium hydride, followed by incorporation of a prototypical MOF, HKUST-1, to obtain composite sheets (MOF@PEO-BTA-Na) that retain the structural integrity of the original 2D polymer sheets. The 2D composite sheets can be assembled into self-supporting membranes and used as a quasi-solid-state electrolyte (QSSE) with a remarkably high ionic conductivity value of 2.80 x 10-3 S cm-1 and a Na+ transference number of 0.95. Consequently, the QSSE facilitates uniform Na plating in Na//Na and Na//Cu cells. Na//NaTi2(PO4)3 cells containing MOF@PEO-BTA-Na QSSE exhibit a high initial specific capacity (129.1 mAh g-1 at 0.5 C), superior rate capability (60.0 mAh g-1 at 20 C), and high-capacity retention (92% after 1000 cycles at 1 C). This work establishes a new, scalable approach for synthesizing 2D organic sheets with promising applications in energy-related areas.