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Carefully transfer the mixture to a blender and blend until smooth. Return the mixture to the pan over high heat. Add the chocolate and cook until reduced to a sauce consistency, about 10 minutes. Season with salt and pepper.
Preheat the oven to 350 degrees F.
Pour the mole into a medium, high-sided saute pan and bring to a simmer. Crack the eggs one a time into a ramekin and carefully slip into the mole. Sprinkle the tops of the eggs with salt and pepper. Transfer the pan to the oven and cook until the whites are just set and the yolks are still jiggly, about 12 minutes.
Spoon some of the Creamy Green Rice into a shallow bowl and make an indention in the center. Put the egg in the center and spoon some of the mole over the egg. Sprinkle with the cheeses and garnish with cilantro leaves.
While the cream is reducing, bring 2 3/4 cups water to a boil in a medium saucepan over high heat. Stir in the rice and season with salt. Bring to a boil again, stir, place a lid on the pan, reduce the heat to medium and cook until all of the liquid is absorbed, 15 to 18 minutes.
Remove the rice from the heat and let sit for 5 minutes with the lid on. Remove the lid and fluff with a fork. Return the rice to the stove over medium heat, add the reduced cream and season with salt and pepper. Stir in the cilantro, green onions and poblanos.