Combine greens, walnut oil, vinegar, 1/2 teaspoon pepper, salt, peas, and cheese in a food processor; process until smooth.
Heat a large nonstick skillet over medium-high heat. Add olive oil to pan; swirl to coat. Crack eggs into skillet. Reduce heat to medium; cook 4 minutes or until whites are set. Sprinkle eggs with remaining 1/4 teaspoon pepper.
Top each bread slice with about 1/4 cup pesto and 1 egg.