Pat chicken dry and remove tenderloin from breasts. Butterfly each breast, then separate butterflied halves to create 4-6 thin pieces; season both sides of chicken with pepper, Italian seasoning, and 1 ½ teaspoons salt.
Pour flour into a shallow dish. Dredge each piece of chicken in flour, coating both sides, and tap off any excess.
In a large skillet over medium-high heat, heat 1 tablespoon oil, then melt 1 tablespoon butter. Add half of chicken and cook, turning occasionally, until golden brown, 2 to 3 minutes per side. Transfer to a plate. Repeat with 1 tablespoon oil and remaining chicken.
In same skillet over medium heat, heat 1 tablespoon oil. Add mushrooms and cook, stirring occasionally, until softened, about 5 minutes. Add shallot and remaining 1 tablespoon oil and ½ teaspoon salt and cook, stirring frequently, until translucent, about 2 minutes.
Add garlic and thyme and cook, stirring, until fragrant, about 1 minute.
Add wine and cook, stirring occasionally and scraping browned bits from bottom of pan, until reduced by half, 4 to 5 minutes.
Add broth and cream and bring to a simmer. Cook, stirring occasionally, until sauce starts to thicken, about 15 minutes.
Stir in remaining 2 tablespoon butter until melted. Return chicken to skillet spoon sauce over chicken until heated through, 2 to 3 minutes. Top with parsley.
