Nonstick cooking oil spray or more olive oil, for greasing
Place a rack in the middle of the oven and preheat to 425° F. Slice the peeled potatoes into ¾ inch pieces.
Place the potato pieces into a gallon-sized plastic zipper bag with olive oil. Close the bag and toss the potatoes well. Work the potatoes with your fingers for a minute so all the potato pieces are lightly covered with oil.
Open the bag and sprinkle in the garlic, paprika, ½ tsp salt and a pinch of black pepper. Close the bag, then shake and massage the potato pieces for a minute or two so that all the pieces are evenly coated with seasoning.
Generously grease a baking sheet with nonstick cooking spray or olive oil. Spread potatoes out in an even layer onto the sheet. Place the baking sheet into the preheated oven and let the potatoes roast for about 20 minutes, stirring once halfway through cooking, till the potato pieces are browned, crisp and cooked through. Reduce oven temperature to 375° F.
While potatoes are cooking, clean the broccoli and cut into small florets. In a medium saucepan, bring 1/2 inch water to a boil. Place broccoli florets in the pan and cover with a lid. Let the broccoli steam for 3-4 minutes till bright green and tender. Pour broccoli into a colander and rinse with cold water to stop the cooking process. Shake broccoli, then pat dry to remove excess water. Rinse the saucepan.
Place broccoli into a large mixing bowl. Add the roasted potato pieces. Toss to mix. Lightly grease your gratin dish or baking dish with olive oil or nonstick cooking oil spray. Pour the mixed vegetables into the dish.
In the rinsed saucepan, melt 1 tbsp butter. Whisk in 2 tbsp potato starch. Add milk slowly, ¼ cup at time, whisking frequently, till it thickens and begins to bubble around the edges. Whisk in the grated cheddar cheese and stir till melted and smooth. Whisk in ½ tsp salt and ¼ tsp pepper. Pour the cheese sauce evenly across the top of the broccoli and potato pieces.
Place the assembled gratin into the oven and bake for 20-30 minutes till edges begin to brown and the cheese sauce is bubbly.
Remove the gratin from the oven and turn on your broiler. When broiler is hot, place the gratin back in the oven and let it brown under the broiler for 1-2 minutes, watching it carefully, till the top is browned to your liking. Serve hot.
Tip: For a thicker cheese sauce, reduce amount of milk to 1 cup.