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Place squash cut-side down in a microwave-safe dish; add 2 tablespoons water. Microwave, uncovered, on High until the flesh is tender, about 10 minutes. (Alternatively, place squash halves cut-side down on a rimmed baking sheet. Bake in a 400°F oven until the squash is tender, 40 to 50 minutes.)
Meanwhile, combine eggs, sauerkraut, breadcrumbs, kielbasa, scallion whites and pepper in a large bowl. Let the squash cool for about 10 minutes, then use a fork to scrape it from the shells into the bowl with the sauerkraut mixture.
Position rack in upper third of oven; preheat broiler.
Heat oil in a large cast-iron or enamel-coated skillet over medium-high heat. Spread the squash mixture evenly in the skillet. Reduce heat to medium and cook, undisturbed, until browned on the bottom, 5 to 8 minutes. Sprinkle with cheese and the reserved scallion greens. Broil until browned on top, 4 to 6 minutes.