3/4 cup low-sodium vegetable broth, plus additional if needed
1 Tbs. unsalted butter
1 medium onion, finely chopped
1 Tbs. olive oil
4 cups peeled, diced eggplant ( 1/2-inch pieces)
1 medium clove garlic, minced
1 Tbs. all-purpose flour
3/4 cup milk
In large nonstick skillet, combine potatoes and 3/4 cup broth; season lightly with salt to taste. Bring to a boil over medium-high heat, cover partially, reduce heat to medium and cook until potatoes are almost tender and most of liquid has evaporated, about 6 minutes. Transfer to bowl; set aside.
In same skillet, melt butter over medium heat. Add onion and cook, stirring often, until softened, about 5 minutes. Stir in oil, eggplant and 2 tablespoons plus 2 teaspoons water; season lightly with salt to taste. Cover, reduce heat to medium-low and cook until eggplant is tender, stirring occasionally, about 5 minutes.
Add garlic and flour and cook, stirring, 1 minute. Reduce heat to low, stir in milk and cook uncovered, stirring, 2 minutes. Stir in reserved potato mixture. Cover and cook, stirring occasionally, until hash is thick and saucy, 5 to 7 minutes. (If hash sticks to pan, stir in additional broth or water by the tablespoon.) During last minute of cooking, sprinkle cheese over hash if desired. Serve hot, garnished with sprigs of fresh parsley.