For the pesto:
2 cups fresh basil leaves
1/4 cup pepitas, lightly toasted
1 garlic clove
1/4 cup freshly grated Parmesan cheese
1/4 teaspoon kosher salt, plus more to taste (if desired)
1/4 cup olive oil
For the hash:
2 tablespoons olive oil
1 shallot (or 1/4 red onion), finely chopped
1 small potato, cut into 1/4-inch-cubes
1 small zucchini, cut into 1/4-inch-cubes
1 ear of corn, kernels removed
1 clove garlic, finely minced
Kosher salt + freshly ground black pepper, to taste
2 eggs
For the pesto: Add all ingredients except the olive oil to a food processor and pulse until a paste forms. With the motor running, slowly add the olive oil and process just until smooth. Transfer to a small bowl. (Can be made a day in advance: cover with a thin layer of olive oil, press plastic wrap directly to the surface and chill)
Heat a large (12-inch) skillet over medium heat. Add the olive oil and the shallot. Let cook, stirring occasionally, until translucent and fragrant, about 3 to 5 minutes. Add the cubed potato and 1/4 cup water. Cover the skillet and reduce the heat to low. Cook for about 7 to 8 minutes, or until potatoes are tender when pierced with a fork (but not falling apart). Remove the cover and add the zucchini, corn, and garlic. Sauté, stirring only very occasionally, until the corn and zucchini is cooked and everything starts to caramelize in places, about 5 to 7 minutes. Season to taste with salt and freshly ground black pepper.
With a spoon, make two small wells in the middle of the hash. Crack the eggs into the wells and cover. Cook until the eggs are finished to your liking, about 5 to 7 minutes for set whites and slightly runny yolks.