This casserole is a riff on a Swiss rosti. It tastes like a loaded baked potato and has the crispness of hash browns.
In a medium pot, bring potatoes to a boil in salted water; cook 10 minutes. Drain; let cool completely. Combine cheese, Dijon, yogurt, and scallions in a large bowl.
Peel potatoes; grate into large shreds (about 6 cups). Fold potatoes, broccoli, eggs, 2 teaspoons salt, and 1/2 teaspoon pepper into cheese mixture. Cover with plastic; refrigerate at least 2 hours and up to 1 day.
Preheat oven to 400 degrees with a rack in top third. In a large oven-proof skillet, cook bacon over medium-high until crisp, about 8 minutes; transfer to paper towels to drain. Add butter to skillet; when it foams, add potato mixture. Press to form an even layer and cook, scraping sides with a spatula to prevent burning, until it starts to set, about 8 minutes. Transfer to oven; bake until golden, 22 to 25 minutes. Let cool slightly. Crumble bacon on top and serve, with sliced scallions and yogurt.