Bring large pot of water to boiling. Peel potatoes; thinly slice; place slices in bowl of cold water as you slice. Drop drained slices into boiling water; boil 3 minutes. Drain; reserve.
Heat oven to 375 degrees F. Coat inside of shallow 12 x 7 x 2-inch or other 2-1/2- to 3-quart baking dish with nonstick cooking spray.
Stir together 1/2 cup milk and the flour in medium-size saucepan until smooth. Stir in remaining 1 cup milk, the half-and-half, water, vegetable broth powder, thyme, salt and pepper; cook, stirring, over medium heat until sauce thickens and bubbles, about 3 minutes.
Spoon one-third of potatoes into prepared casserole. Sprinkle with half of the ham and green onion. Spoon on one-third of sauce. Repeat layers. Top layer should be potatoes with remaining sauce poured over all.
Bake, covered, in 375 degree F oven for 30 minutes. Sprinkle cheese over top and dust with paprika if desired. Bake, uncovered, 15 minutes or until lightly browned. Let stand 10 minutes before serving. Makes 6 servings.