Preheat your oven to 375°F and lightly grease a small baking dish.
Place the poblano pepper under the broiler, turning occasionally until the skin is charred and blistered on all sides.
Place the charred pepper in a sealed bag for 5 minutes to steam, then peel off the skin, remove the seeds, and roughly chop.
Cook the whole wheat spaghetti in a pot of boiling salted water until al dente, then drain and set aside.
Season the chicken breast with a pinch of salt and pepper, then sauté in a pan with olive oil over medium heat until cooked through; shred the chicken using two forks.
In a high-speed blender, combine the roasted poblano, Greek yogurt, chicken broth, garlic, cilantro, cumin, and remaining salt and pepper until the sauce is completely smooth.
In a large bowl, toss the cooked spaghetti and shredded chicken with the green poblano sauce until evenly coated.
Transfer the mixture to the prepared baking dish and sprinkle the shredded mozzarella evenly over the top.
Bake for 15 minutes or until the sauce is bubbling and the cheese has melted into a light golden crust.