Step 1: Pat the wild turkey breast thoroughly dry with paper towels. Season generously on both sides with salt and freshly ground black pepper. If the breast is very thick (more than 1.5 inches), consider pounding it slightly to an even 1-inch thickness for quicker, more uniform cooking.
Step 2: Heat 1 tablespoon of olive oil and 1 tablespoon of butter in a large skillet or cast-iron pan over medium-high heat. Once the butter is melted and sizzling, carefully place the turkey breast in the pan.
Step 3: Sear the turkey for 5-7 minutes per side, or until deeply golden brown and an instant-read thermometer inserted into the thickest part reads 160°F (71°C). The internal temperature will rise to 165°F (74°C) as it rests. Remove the turkey from the pan and set aside on a cutting board to rest, loosely tented with foil.
Step 4: Reduce the heat to medium. Add the remaining 1 tablespoon of butter to the same skillet. Add the sliced mushrooms and sauté for 5-7 minutes, stirring occasionally, until they are golden brown and have released their liquid. Scrape up any browned bits from the bottom of the pan as you stir.
Step 5: Add the minced garlic, chopped fresh thyme, and rosemary to the pan. Cook for another 1 minute until fragrant. Be careful not to burn the garlic.
Step 6: Pour in the chicken broth, scraping the bottom of the pan with a wooden spoon to deglaze and incorporate all the flavorful browned bits. Bring to a simmer and cook for 2-3 minutes, allowing the sauce to reduce slightly. If using, stir in the heavy cream and simmer for an additional 1-2 minutes until slightly thickened.
Step 7: Taste the mushroom sauce and adjust seasoning with salt and pepper as needed. Slice the rested wild turkey breast against the grain into thick medallions. Serve immediately, drizzling generously with the savory mushroom sauce and garnishing with fresh chopped parsley.