1. Bring a large pot of water to a boil over high heat. Cook pasta according to package directions; reserve 1/4 cup of pasta cooking water. Add chard after 5 minutes of cooking and drain; return pasta and greens to pot.
2. While pasta is cooking, combine parsley, spinach, walnuts, garlic, 2 tablespoons oil, lemon zest, 1 tablespoon water, and 1/4 teaspoon each salt and pepper in a food processor; pulse until chunky.
3. In a large nonstick skillet, heat remaining 2 teaspoons olive oil over medium-high heat. Working in batches, crack eggs into skillet. Cook until the whites are completely firm but the yolks are still soft (about 2 minutes). Season eggs with remaining 1/4 teaspoon each salt and pepper.
4. Toss reserved pasta mixture with pesto and enough reserved cooking water to make a thin sauce. Divide pasta among 6 shallow bowls and top each with a fried egg.