To begin, prep all of your vegetables - mince the garlic, chop the shallot, and trim off any stems from the haricots verts. Trim off the woody ends from the asparagus, then cut them a into thirds. You can also cut some of your snap peas open, but that's optional!
Add the olive oil to a large, lidded pan, and bring to medium-low heat. Add in the garlic and shallot, and sauté until fragrant, about 2-3 minutes. Stir often enough so the garlic doesn't burn.
Then, add in the haricots verts, asparagus, and snap peas to the pan. Stir around for another 1-2 minutes. Add this point, add in the salt and pepper, and stir well.
Then, deglaze the pan with the white wine. (If you'd prefer not to add white wine, you can skip this step.) Stir everything around in the white wine and cook for about 1-2 minutes.
Add your pasta to the pan. Hopefully your pan is large enough to accommodate the pasta without having to break it. But if you have to break the pasta in order for it to fit, that's okay! Just don't tell any Italians! ;)
Add in enough water to just cover the pasta. You can use room temperature water, though it will slow down the cooking process a little. If you're in a rush, you can use boiling water instead. Depending on your pan, you should be adding between 3-4 cups of water.
Stir the pasta in the water a little so that each piece is submerged in the water. Then, bring the heat up to high. Cover your pot, and let the pasta cook for about 5 minutes. After 5 minutes, you can uncover the pan and stir the pasta around. If the pasta is still very hard but it looks like the water level is getting to low, add in more water in 1/4 cup increments, then re-cover the pan. Keep doing this until the pasta is cooked at al dente.
Check your box of spaghetti for a reference on the cooking time. You want there to be a little bit of liquid left in the pan when the pasta is al dente, because it will create a creamy sauce with the cheese. However, you don't want too much liquid or the pasta will be soupy.
Remove the pan from heat, then squeeze in the juice from 1/2 a lemon and stir (you can add more lemon juice if you'd like!) Then, stir in the parmesan cheese until everything is coated and the "sauce" is creamy. After this, stir in the arugula.
Plate the pasta in bowls with extra arugula for garnish, and with a little more freshly-shaved parmesan on top. Enjoy!