A milling cutter for sharpening welding electrodes includes a tool body, and a plurality of cutting inserts fastened to the tool body. The inserts include a center cutting insert and at least two peripheral cutting inserts. Each peripheral cutting insert includes at least one cutting edge. The center cutting insert defines: two oppositely facing principle planes, two cutting edges disposed adjacent respective ones of the principle planes, and a concave surface extending between such cutting edges. The center cutting insert can cut (sharpen) the end surfaces of two welding electrodes simultaneously.