In a food processor, combine the pea shoots with the mint, walnuts, lemon juice, and garlic and process until roughly chopped. With the motor running, slowly drizzle in olive oil until emulsified and smooth. If 1/3 cup isn't enough, add a little more.
Scrape the pesto into a bowl and season with salt. Use immediately, or store in an airtight container, covered with a thin layer of olive oil.