Season chicken on both sides with salt and pepper. In a large pan over medium-high heat, heat 1 tablespoon oil. Add chicken and cook until golden, 4 minutes per side. Remove from skillet.

Reduce heat to medium and add remaining 1 tablespoon oil. Add shallot and mushrooms and cook until golden, 5 minutes.
Add garlic, thyme, and oregano and cook until fragrant, 1 minute more.
Add broth and scrape up any brown bits on bottom of pan, then slowly add in half and half. Bring to a simmer and season with salt, pepper, and a pinch of red pepper flakes.
Stir in gnocchi and return chicken to pan. Let simmer until chicken is cooked through and internal temperature reaches 165°, 8 to 10 mins

Add cheeses and stir until melted, then add spinach and stir until wilted.

Slice chicken and return to skillet. Season with more salt and pepper to taste.