Prepare filling: In oven-safe nonstick 10-inch skillet (if skillet is not oven-safe, wrap handle with double layer of foil), heat 2 teaspoons oil over medium heat until hot. Add red pepper and cook, stirring occasionally, until tender and lightly browned, 10 to 12 minutes. Transfer pepper to small bowl; set aside. Wipe skillet with paper towel.
Meanwhile, prepare potato pancake: Preheat oven to 400 degrees. Peel and coarsely shred potatoes; pat dry with paper towels. In large bowl, toss potatoes with salt and black pepper.
In same skillet, heat 1 tablespoon oil over medium heat. Working quickly, add half of potatoes, gently patting with rubber spatula to cover bottom of skillet. Leaving 1-inch border, top potatoes with red pepper and goat cheese. Cover filling with remaining potatoes, gently patting to edge of skillet. Cook, gently shaking skillet occasionally to keep pancake from sticking, until browned, 10 minutes.
Carefully invert pancake onto large, flat plate. Add remaining 1 tablespoon oil to skillet, then slide pancake back into skillet. Cook, gently shaking skillet occasionally, 10 minutes longer.
Place skillet, uncovered, in oven and bake until potatoes are tender throughout, 20 to 25 minutes.