This article presents the design and optimization of a miniaturized 48-V-to-1-V, 240-A linear-extendable group operated point-of-load (LEGO-PoL) CPU voltage regulator module (VRM)— Mini-LEGO —with a volume of 30 mm×11.2 mm × 8.4 mm. The Mini-LEGO converter provides vertical power delivery, and it achieves a peak efficiency of 87.1%, a full load efficiency of 84.1%, and a power density of 1390 W/in$^{3}$ when stepping down a 48 V input voltage to an output voltage of 1 V. The Mini-LEGO offers compact device implementation, simple gate drive circuitry, and a fast regulation-stage switching frequency of 1.5 MHz. The 9-mm-by-9-mm, 4-phase, 20-A-per-phase vertical coupled inductor has a per-phase transient inductance of 10.3 nH, with its coupling reducing the ripple by a factor of six, and a height of 2.5 mm. This vertical coupled inductor structure is designed following a systematic approach to minimizing its height while achieving significant current ripple reduction and maintaining sufficient saturation margin. Thermal simulations are performed to verify the cooling feasibility. A measurement characterization method for vertical coupled inductor structures is introduced and used to characterize the designed vertical coupled inductor.