Bring a pot of salted water to a boil and cook the chickpea pasta according to package directions, reserving two tablespoons of the cooking water before draining.
While the pasta cooks, cut the chicken breast into bite-sized pieces and season with sea salt and black pepper.
Heat the olive oil in a large skillet over medium-high heat and sauté the chicken until golden and cooked through, approximately 5-6 minutes.
Add the sliced cremini mushrooms and minced garlic to the skillet, cooking until the mushrooms have released their moisture and are browned.
In a small bowl, whisk together the Greek yogurt, nutritional yeast, fresh thyme, and the reserved pasta water until smooth.
Reduce the skillet heat to low, add the cooked pasta, and pour the yogurt mixture over the top, tossing gently to coat without boiling the sauce.
Garnish with fresh chopped parsley and serve immediately while warm.