Prepare pesto: Bring one gallon salinated water to a rapid boil. Blanch and shock basil leaves by adding them to boiling water for about 5 seconds, then quickly submerging them in an ice bath. (This will set the chlorophyl in the basil so the pesto remains bright green.) Squeeze excess water out of basil and roughly chop.
In a blender or food processor, process basil with enough oil to emulsify (about 3 to 4 tbsp). Add garlic, nuts and cheese and season with salt. Continue blending just enough to emulsify. (Pesto should have a slightly course texture.) Chill until needed. Pesto will keep up to 2 days but is best served the day that it is made.
Prepare pasta: Preheat oven to 350°F In an oven safe baking dish, arrange tomatoes cut side up. Arrange thyme sprigs over top, season with salt and pepper, and drizzle with oil. Bake tomatoes until they just begin to brown and caramelize, approximately 15 minutes.
Bring 1 gallon salinated water to a rapid boil. Meanwhile, mince potato either by hand or in a food processor. Blanch beans in boiling water for about 4 minutes, then quickly submerge in an ice bath. Set aside. In the same water, add pasta and potato and cook until pasta is tender. (Cooking time will vary by pasta type used, but standard time is 7 to 9 minutes.) Strain pasta along with potato.
In a large mixing bowl, combine tomatoes, pesto, beans and pasta-potato mixture. Toss until well combined.