How to Make It
Preheat broiler to high with rack 8 inches from heat. Place potatoes and 2 tablespoons water in a microwavable bowl; cover with plastic wrap. Microwave on high until tender, 7 to 8 minutes. Transfer potatoes to a work surface; let cool for 5 minutes. Gently flatten potatoes with a small bowl. Transfer to a rimmed baking sheet; drizzle with 2 tablespoons oil, turn to coat, and arrange in an even layer. Broil until light brown, about 6 minutes.
Toss broccoli with 2 tablespoons oil and add to baking sheet in an even layer alongside potatoes. Season broccoli and potatoes with pepper and ¾ teaspoon salt. Return to oven; broil, watching carefully, until broccoli is tender and potatoes are golden brown, 10 to 12 minutes.
Meanwhile, heat remaining 1 tablespoon oil in a large skillet over medium-high. Fry eggs until whites are set and yolks are still runny, 3 minutes. Serve broccoli and potatoes topped with cheese, eggs, dill, and remaining ¼ teaspoon salt.