Place a medium pot over medium-high heat with 1 turn of the pan of EVOO, about 1 tablespoon. Add in the rice and cook about a minute (you're toasting it, so it's ready when it starts to smell nutty). Add in 2 1/2 cups stock and bring up to a bubble. Turn the heat down to medium-low and simmer the rice until tender, about 20 minutes.
Make the "green" part of the rice by puréeing the baby spinach, scallions, garlic, ginger, shiso or cilantro, lime juice and remaining stock (about 1/2 cup) together in a food processor. Reserve it until the rice is finished cooking.