Bring one large pot and a medium saucepan of salted water to a boil. Cook the linguine in the large pot according to package directions. Drain and reserve.
In the small pot, cook the green beans for 2 minutes. Then add the carrots and cook for about 5 more minutes, or until the vegetables are crisp tender. Drain and reserve.
Heat a medium saucepan over medium heat. Add 2 TBSP water and the shrimp. Sprinkle with salt and pepper.
As soon as the shrimp start to turn pink, turn them over and cook another 30 seconds (there should be a little water still left – if not, add a touch more).
Add the pasta and reserved vegetables and toss with the pesto. Heat through and divide between 4 bowls.