Preheat your oven to 400°F and line a baking sheet with parchment paper.
Toss the sliced cremini mushrooms with 0.5 tsp of olive oil and a pinch of salt and pepper, then roast for 15 minutes until golden brown.
Bring a large pot of salted water to a boil and cook the tagliatelle until al dente, reserving 0.25 cup of the pasta water before draining.
Season the chicken breast with the remaining salt and pepper, then sear in a skillet with the remaining 0.5 tsp olive oil over medium-high heat for 6 minutes per side.
Remove the chicken from the pan, let it rest for 3 minutes, then slice into thin strips.
Turn the skillet heat to low, add the minced garlic, and sauté for 30 seconds until fragrant.
In a small bowl, whisk together the Greek yogurt, grated parmesan, and the reserved pasta water until a smooth sauce forms.
Add the cooked pasta, roasted mushrooms, and chicken strips back into the skillet, pour the yogurt sauce over the top, and toss gently until the sauce is creamy and well-distributed.
Garnish with chopped fresh parsley and serve immediately.