Cut chicken into bite-sized pieces; season with salt and pepper.
Heat oil in a large skillet over medium. Add chicken and chopped mushrooms. Allow to cook for 5-6 minutes, stirring to brown all sides. Set aside covered.
Add broth, salsa verde, milk and pasta to skillet. Boil, cover, and reduce heat to low; simmer for 15 minutes, stirring occasionally.
Check pasta to be sure it has finished cooking, then add chicken and mushrooms back to skillet. Stir in the cheese; cover and cook for 2 minutes, allowing cheese to melt. Remove from heat; stir in spinach and allow to wilt for 1 minute. Serve with grapes on the side.