Swiss rösti are traditionally made with only potatoes, but you can use other veggies, too, particularly parts of ones you may typically discard: cabbage and cauliflower cores, broccoli stems and the ribs of dark leafy greens. Poached eggs give the dish staying power.
Toss grated and/or chopped vegetables with salt in a large bowl. Pile in the center of a clean kitchen towel, gather up the ends and twist them together, squeezing to remove as much excess moisture as possible. Whisk 2 eggs in the bowl, then stir in the squeezed vegetables.
Grate potatoes on the large holes of a box grater and place in another large bowl. Add enough cold water to cover by several inches and swish to release excess starch from the potatoes. Drain. Twist the shredded potatoes in the towel and squeeze out all the excess moisture. Add to the bowl with the vegetables. Stir in flour, scallions, granulated garlic and pepper.
Heat 2 tablespoons oil in a large nonstick or cast-iron skillet over medium-high heat. Using 1/2 cup of the vegetable mixture for each, spoon 4 portions into the pan, flattening to 3 1/2- to 4-inch cakes. Reduce heat to medium-low and cook, flipping once, until golden brown, about 4 minutes per side. Transfer the rösti to a baking sheet and keep warm in the oven. Repeat with the remaining 2 tablespoons oil and vegetable mixture.
To poach the remaining 4 eggs, break each into a small bowl. Fill a large, straight-sided skillet or large pot with 2 inches of water; bring to a boil. Add vinegar. Reduce to a gentle simmer: the water should be steaming and small bubbles should come up from the bottom. Submerging the lip of each bowl into the simmering water, gently add the eggs, one at a time. Cook for 4 minutes for soft-set, 5 minutes for medium-set and 8 minutes for hard-set. Using a slotted spoon, transfer the eggs to a clean kitchen towel to drain for a minute.
Serve the poached eggs on the rösti.