To prepare potatoes, place potatoes in a large saucepan; cover with water. Bring to a boil. Reduce heat; simmer 15 minutes or until tender. Drain.
Return potatoes to pan; add 2 tablespoons oil, 1/2 teaspoon salt, and 1/4 teaspoon pepper. Mash mixture with a potato masher to desired consistency. Cover and keep warm.
Heat milk over medium heat in a large nonstick skillet to 180° or until tiny bubbles form around edge (do not boil). Stir in spinach; cook 1 minute or until spinach wilts. Add milk mixture and green onions to potato mixture, stirring until well combined.
To prepare ragout, wipe skillet with paper towels. Heat 2 teaspoons oil in skillet over medium heat. Add sliced onion, oregano, and garlic; cook 5 minutes or until lightly browned, stirring frequently. Add mushrooms; cook 7 minutes or until mushrooms are tender, stirring frequently. Add zucchini; cook 2 minutes, stirring frequently.
Stir in water and next 5 ingredients (water through tomatoes); simmer, uncovered, 15 minutes, stirring occasionally. Serve ragout over potatoes, and sprinkle with cheese.