Heat 1 tablespoon olive oil in a large, high-sided, heavy-bottomed skillet. Cook sliced mushrooms, generously seasoned with salt and pepper, on medium-high heat for about 1 or 2 minutes until lightly browned and softened. Remove half of the cooked mushrooms from the skillet to a plate.
To the same skillet with the half of cooked mushrooms, add uncooked orzo, chicken broth, minced garlic, Italian seasoning, smoked paprika, and ¼ teaspoon salt.
Bring to a boil, then reduce to a simmer (medium-low) and cook for about 5 or 10 minutes, occasionally stirring to prevent the orzo from sticking to the bottom of the pan, until the orzo is cooked through.
Add fresh spinach during the last 5 minutes of cooking the orzo.
Add ½ cup of cream. Stir everything on low-medium - do not bring the sauce to a boil.
Season with salt and more smoked paprika, if desired. Stir in the remaining half of the cooked mushrooms.
When serving, sprinkle a small amount of red pepper flakes over the creamy orzo.