Melt butter in 5-quart Dutch oven over medium-high. Cook mushrooms and salt 8 minutes, stirring occasionally, until liquid evaporates and some edges brown.
Cook shallot 2 minutes; add garlic and thyme 30 seconds.
Add broth and milk; gently simmer. Add gnocchi; cover and cook over medium-low 4 minutes.
Uncover; carefully stir and add spinach and pepper. Cook 2 to 3 minutes until spinach wilted and gnocchi tender throughout, adding reserved water only if sauce dries.
Remove from heat; fold in Parmesan. Rest 2 minutes; serve in 4 bowls while sauce still flows between gnocchi.