In a food processor, pulse eggs, olive oil, flour, salt, and water together until a crumbly mixture forms. Test to make sure it is the correct consistency by pressing together a handful to see if it forms a dough ball. Add more water or flour to adjust, if needed.
Turn the mixture out onto a lightly floured surface and knead together along with the herbs. Knead for a few minutes until it is smooth.
Wrap in plastic or a damp towel and allow to rest for 30 minutes.
Divide the dough in half or into smaller, more manageable pieces and roll out into rectangles. You may also use a pasta machine. Roll to the lowest setting in the machine, passing the dough through twice at each setting. Or, roll out as thin as you can with a rolling pin.
Layer the sheets of dough on a sheet pan with flour between the layers. Top with a damp towel while you roll out all of the dough.
Cut the sheets into ¾ inch wide strips. Let them dry out as you continue cutting.
Bring a pot of salted water to a boil. Blanch asparagus and peas until tender and bright green.
Reserve the same water and bring it to a boil again to cook the pasta.
Melt butter in a medium-hot saucepan and cook the leeks.
Toss in the asparagus and peas. Remove from heat.
Cook pasta for about 3 minutes or until al dente.
Toss pasta in olive oil along with arugula and then combine with the vegetable-butter mixture.
Garnish with lemon zest, parmesan, and additional fresh herbs, if desired. Taste and adjust salt and pepper.