Place the poblano peppers under a broiler or over an open flame, turning occasionally until the skin is charred and blistered on all sides.
Place the charred peppers in a bowl and cover with a lid or plastic wrap for 10 minutes to steam, then peel off the skin, remove the seeds, and roughly chop.
Bring a large pot of salted water to a boil and cook the chickpea spaghetti according to package directions until al dente.
While the pasta cooks, season the chicken breast with half of the salt and pepper, then grill in a pan over medium-high heat until fully cooked through, about 5-6 minutes per side.
In a blender, combine the roasted poblanos, Greek yogurt, cilantro, garlic, white onion, and the remaining salt and pepper. Blend until completely smooth.
Heat the olive oil in a large skillet over medium heat. Pour in the poblano sauce and simmer gently for 3-4 minutes to meld the flavors.
Drain the pasta and add it directly to the skillet with the sauce, tossing well to coat every strand.
Slice the grilled chicken into strips and serve on top of the creamy green spaghetti.