Make the pesto: Place basil, parsley, and mint leaves in a medium bowl; add 4 cups ice and enough cold water to submerge. Let stand 10 minutes.
Combine extra-virgin olive oil and garlic in a food processor, and pulse until garlic is chopped. Reserve 2 tablespoons ice water; drain herbs. With food processor running, add herbs through food chute; process until finely chopped, stopping to scrape down sides of bowl as necessary. Add feta, frozen peas, 1 teaspoon sea salt, and reserved 2 tablespoons ice water; pulse until creamy. Scrape into a bowl. Fold in sugar snap peas, and season to taste with salt and pepper.
Make the spaetzle: Whisk together flour, 2 tablespoons kosher salt, and pepper in a large bowl. Whisk together milk, eggs, and minced parsley in a separate medium bowl. Add wet ingredients to dry ingredients; whisk until just smooth. Let stand 20 minutes. (The batter should be thicker than pancake batter.)
Meanwhile, grease a 9- by 13-inch baking dish with butter. Bring a large pot of water to a boil over high, and season with remaining 2 tablespoons kosher salt.
If using a spaetzle maker, place spaetzle maker over pot of boiling water. Working in batches, fill hopper with 1 cup batter, and then slowly move hopper back and forth over grate, scraping with a rubber spatula to create 1-inch-long spaetzle. If not using a spaetzle maker, set a colander with 1/4-inch holes over pot of boiling water. Working in batches, spoon 1 cup batter into colander, and use a rubber spatula to press batter through holes, creating 1-inch spaetzle.
Stir spaetzle once or twice to separate. As soon as spaetzle rise to surface, use a slotted spoon to transfer to a clean colander. Rinse with cold water. Shake spaetzle dry. When all spaetzle are cooked, toss with olive oil, and place in prepared baking dish.
Preheat oven to 350°F. Melt 2 tablespoons butter in a small saucepan over medium. Add leeks; stir to coat. Cook, stirring occasionally, until leeks are soft, about 12 minutes. Add leeks to prepared baking dish with spaetzle, and fold to combine.
Make the fonduta: Pour cream into a large saucepan. Bring to a simmer over medium, and cook until reduced by one-third, about 10 minutes. Remove from heat, and whisk in Manchego, egg yolk, and 1 teaspoon kosher salt. Cover saucepan to keep warm.
Bake spaetzle and leeks in preheated oven until spaetzle are crisp and browned, 15 to 20 minutes. Pour fonduta over spaetzle mixture. Dollop 1/2 cup pesto over fonduta. (Reserve remaining pesto for another use.) Garnish with pea tendrils and serve immediately.