Place flour into a large bowl.
Crack eggs into a blender. Add 3 cups green garlic leaves and 1/4 cup water and blend on high speed until very well mixed.
Pour egg and green garlic mixture into the bowl of flour and combine with a chopstick or fork until shaggy. Once dough has started to come together, place it on your work surface.
Knead into a cohesive, elastic dough for 10-15 minutes and cover with plastic wrap. Rest in the fridge for at least 30 minutes, preferably 60 minutes.
Meanwhile, rinse and dry your blender. Then add 4 cups of green garlic leaves + stems, olive oil, walnuts, 2 tbsp lemon juice, parmesan, and 1 tbsp sea salt. Pulse on high speed until chunky but blended.
Pour pesto into a bowl and cover tightly. Place in the fridge to rest until ready to make the sauce.
Remove your dough from the fridge and cut into 4 equal pieces, placing 3 back in the plastic wrap, completely covered, to rest while you roll out your dough.
Lightly flour your work surface. Press your first pieces of dough into a flat, long rectangle shape. Roll through your thickest pasta roller setting.
Continue rolling your pasta through thinner and thinner settings until you reach setting “6”. If you are rolling by hand, roll until you can see your hand through the pasta sheet.
With your long and thin pasta, trim sides to have even flat edges. Then, cut your pasta into even squares (about 2" x 2").
Place each square, one at a time, on your gnocchi board, pointy side up. Using a chopstick or thin wooden dowel, wrap the bottom edge around the dowel to secure and roll up, pressing the dough into the board as you form tube shapes.
Repeat this process of rolling dough and shaping squares then rolling into garganelli, allowing your garganelli to dry on a lightly floured plate.
Once you’re ready to cook, heat a large pasta pot with 6-8 quarts of water over high heat and add a small palmful of table salt.
While your water heats, chop the rest of your green garlic and sauté them in 2 tbsp of butter on a gentle simmer in a large sauté pan.
Boil 1/6 of your pasta at a time, about 24-32 garganelli per person, until it floats on top of the pasta pot. Then, hoist your pasta into the sauté pan of butter and green garlic. Shake to combine.
Add 1/4 cup pasta water and a heaping spoonful of your green garlic pesto. Shake the pan to combine and cook for 1-2 minutes.
Plate your pasta and sauce and garnish with a sprinkle of sea salt for flavor and an extra squeeze of lemon juice. Repeat for all 6 portions of garganelli.