Using a box grater or a food processor with the shredding disk, shred the zucchini, sweet potato and parsnip; transfer to a colander. Toss with 1 tsp. salt and let stand for 20 minutes. Peel and shred the baking potato, roll it in a towel and squeeze out the excess liquid; transfer to a bowl. In the towel, roll up handfuls of zucchini mixture and squeeze to remove the excess liquid. Add to the baking potato. Add the pastrami, 2 tbsp. EVOO, 1/2 tsp. salt and 1/4 tsp. pepper; toss.
In a large nonstick skillet, heat 2 tbsp. EVOO over medium-high heat. Spread the hash mixture evenly in the pan, lower the heat to medium and cook, undisturbed, until hash is golden-brown on the bottom, about 10 minutes. Using a heatproof spatula, divide the hash into sections, then turn and cook until golden outside and tender inside, about 5 minutes. Divide the hash among plates.
In another nonstick skillet, heat 2 tbsp. EVOO over medium heat. Fry the eggs 3 or 4 at a time to the desired doneness, adding the remaining 2 tbsp. EVOO if needed. Top the hash with the eggs.