By vegetarian do you mean vegan (no dairy)?
I make a scalloped potato dish that is very yummy but it does have milk.
I don't have an exact recipe - I just learned from my grandma.
Choose a casserole dish the size you want and then peel and thinly slice enough potatoes to layer and fill the dish about 3/4 of the way up (but don't layer them all at once). Peel and slice ahead of time and then keep them in a bowl of cold water to keep them from browning. Chop up a small to medium onion (depending on how much you like onion
)
In a pot melt 2 tablespoons butter on medium heat and add about 1/2 teaspoon of salt and some ground pepper to taste. Add 2 tablespoons of flour to make a paste. Slowly add 2 cups milk (you can use any kind of milk - I use skim and it works fine but whole or 2% will make it richer). Whisk constantly over medium heat until sauce thickens into a roux (the consistency of gravy).
Butter casserole dish and start by layering potatoes - I usually make about 1-inch layers. In between each layer pour some of the sauce and sprinkle some onions. Keep alternating potatoes in 1-inch layers and sauce, ending with sauce on top. You can also season each layer with salt, pepper, and garlic powder if you like (we love garlic).
Cover with foil and bake at 350 for 45 minutes then uncover and bake another 15 minutes. When they are done a fork should be able to go right through the layers easily.