Heat a Dutch oven over medium-high heat with olive oil. Add sliced cremini mushrooms in an even layer. Do not stir for 3-4 minutes to achieve a deep brown crust. Toss, season with salt, and remove to a plate.

Lower heat to medium, melt butter in the pot. Sauté diced onion for 4-5 minutes until translucent, then add minced garlic for 60 seconds. Sprinkle flour over the top and whisk constantly for 2 minutes to form a pale golden paste.

Slowly pour in the chicken broth while whisking vigorously to prevent lumps. Submerge the raw chicken breast into the liquid. Bring to a gentle simmer, cover, and cook for 15-18 minutes until the chicken reaches 165°F (74°C).

Remove the cooked chicken and shred it into bite-sized chunks using two forks. Return the shredded chicken and deeply browned mushrooms to the pot.

Reduce heat to low and stir in the heavy cream. Warm through for 2 minutes until it becomes a velvety off-white color. Ladle into shallow bowls, drizzle with a drop of olive oil, and garnish with chopped parsley and coarse black pepper.