Clean potatoes, rub with a little olive oil, and sprinkle with salt. Pierce potatoes with a fork, wrap in foil, and place on a baking sheet. Bake until slightly tender, about 45 minutes.
Allow potatoes to cool until they can be handled. Cut potatoes into 1-inch cubes. Toss together cut potatoes, broccoli, half of each cheese, pepper, and salt, and add to a 9 x 13-inch casserole pan.
In a medium saucepan, melt butter over medium-low heat. Add flour and whisk until smooth.
Slowly pour in the broth, raise to medium heat, and whisk until it reaches a gravy-like consistency. Remove from heat and add yogurt and green chiles.
Pour mixture over potatoes and broccoli and top with remaining mozzarella, then sprinkle cheddar in rows for a mouthwatering effect. Cover with foil and bake 20 minutes. Remove foil and bake an additional 10 minutes, or until cheese is melted and bubbly.