In a strainer placed in the sink or over a plate, combine the potatoes and zucchini. Squeeze to remove excess water. Add 3 tablespoons of the dill and toss.
Heat the oil in a large cast-iron skillet over medium-low heat.
Add the potato-zucchini mixture, pressing it with the back of a spatula. Season with the salt and pepper. Cook, without stirring, for 15 minutes.
Fold the hash in half. Cook for a few more minutes.
Transfer to individual plates along with the eggs. Sprinkle the hash and eggs with the remaining dill.