Light a grill or preheat the broiler. Roast the poblanos over the grill or under the broiler, turning, until blackened, about 3-4 minutes. Transfer the chiles to a bowl and cover with plastic wrap until they cool. Peel, core and seed the poblanos then finely chop them.
Brush the portobello mushrooms with 3 tablespoons of the olive oil and season with salt and pepper. Grill or broil the mushrooms over high heat, turning, until softened, about 10-12 minutes. Transfer the mushrooms to a plate, stem side down, to let them drain and cool.
Meanwhile, in a medium skillet, heat the remaining teaspoon of olive oil and add the onion. Cook over moderate heat until softened, about 6 minutes. Add the baby spinach and cook until the leaves wilt, about 1 minute. Transfer the spinach to a sieve and press out the liquid.
In a bowl, mix the spinach with the rice, cheese, cilantro and poblanos and season with salt and pepper.
Preheat the oven to 325°F.
Season the mushroom caps with salt and pepper and spoon some of the rice mixture into the mushrooms, mounding it slightly. Transfer the mushrooms to a baking dish and bake for about 20 minutes, or until the cheese is melted and the tops are lightly browned.