Line a large bowl with a clean kitchen towel and line a baking sheet with two layers of paper towels.
Place the grated potatoes in the towel then squeeze out all of the liquid from the potatoes. Discard the liquid then place the dried potatoes back in the bowl.
Stir in the egg, bread crumbs, salt and pepper until combined.
In a heavy saucepan, heat 1 tablespoon of butter and 1 tablespoon of olive oil over medium-low heat. To test if the oil/butter mixture is hot enough, drop a small piece of the potato mixture into the pan and if bubbles form around the edges, the oil is ready.
Carefully place two ¼ cup-sized scoops of the potato mixture in the pan and cook for two minutes, until the latkes are browning on the bottom.
Flip the latkes over carefully with a spatula and place 1-2 slices of cheddar cheese on one latke. Cook for one more minute then place the second latke on top of the cheese and press down. Cook for 30 seconds then flip over the grilled cheese latke sandwich, press down and cook for 30 more seconds.
Place the hot grilled cheese latkes on the paper towels to drain then repeat this process to make 3-4 more sandwiches.
Once the latkes have drained, it's time to eat!
If you want to take these grilled cheese latkes to the next level, you can add avocado or spinach to the cheese while cooking or switch it up with your favorite cheese.
Place everything but the olive oil in a food processor or blender and blend until it forms a thick paste. Then, with the processor on, drizzle in the olive oil. Blend until smooth. (If using a blender, add the olive oil all at once instead of slowly drizzling it in.)
Serve on the side as a dip, or add a dollop to the top of your latkes.