Boil salted water, cook pasta to al dente per package directions
Reserve 1 cup pasta water before draining
Wipe mushrooms clean with damp towel, slice ¼-inch thick
Heat butter and olive oil in large skillet over medium-high
Add mushrooms in single layer, cook undisturbed 3-4 minutes until golden
Stir, cook 3-4 more minutes until tender and browned
Reduce heat to medium, add garlic, cook 1 minute
Add broth, scrape pan bottom, simmer 2-3 minutes until reduced by half
Add cream, stir to combine, bring to gentle simmer
Cook 3-4 minutes until slightly thickened
Stir in thyme, basil, salt, pepper, nutmeg, red pepper flakes
Simmer 2 minutes
Add spinach in batches, stirring as it wilts
Reduce heat to low, add Parmesan, stir until melted
Don't let boil after adding cheese
Add pasta, toss to coat, adding pasta water ¼ cup at a time
Adjust consistency—add water if thick, simmer if thin
Taste and adjust seasoning
Divide among bowls
Top with Parmesan, pepper, parsley
Serve immediately