Pat chicken dry; season with Italian seasoning, salt, and pepper. Place ¼ cup flour in a shallow dish. Dredge chicken on both sides in flour, then transfer to a plate.

In a large Dutch oven or heavy pot over medium-high heat, heat 2 tablespoons oil until shimmering. Working in batches if needed, cook chicken, turning occasionally, until golden brown and an instant-read thermometer inserted into thickest part registers 165°, 5 to 6 minutes per side. Transfer to a cutting board. Let cool, then shred into bite-sized pieces.

In same pot over medium heat, heat remaining 2 tablespoons oil. Cook mushrooms, stirring occasionally, until softened and starting to brown, 10 to 12 minutes; season with salt. Transfer to a plate.

In same pot over medium-high heat, melt butter. Add shallots, garlic, and thyme; season with salt. Cook, stirring frequently, until shallots start to soften and garlic is fragrant, about 3 minutes. Whisk in remaining ¼ cup flour and cook, whisking frequently, until smooth, about 2 minutes. While whisking, slowly stream in wine and cook, whisking, until a thick slurry forms, about 2 minutes. While whisking, slowly stream in broth. Increase heat to high to bring to a boil, then reduce heat to medium and bring to a simmer. Cook, stirring occasionally, until slightly thickened and flavors have melded, about 15 minutes.

Return mushrooms to pot. Add cream and return to a simmer. Add gnocchi and cook, stirring occasionally, until al dente according to package directions.

Stir in chicken and parsley; season with salt, if needed.