Preheat the oven to 375 degrees F. Combine the cilantro sprigs, garlic, chicken broth and 1 1/4 teaspoons salt in a blender and puree. Transfer to a liquid measuring cup and add enough water to make 3 3/4 cups liquid.
Heat the vegetable oil in a medium saucepan over medium-high heat. Add the rice and cook, stirring, 1 minute. Add the cilantro puree and bring to a boil. Reduce the heat to low, cover and cook until the liquid is absorbed, about 20 minutes. Remove from the heat and let stand, covered, 10 minutes.
Spread the plantains on a baking sheet and bake until heated through, 15 minutes. Stir the chopped cilantro into the rice. Top with the plantains.