Preheat the oven to 425°F. Spray a rimmed baking sheet with nonstick spray.
Slice the potatoes into 1-inch thick rounds.
In a large bowl, combine the butter, garlic, thyme, salt and pepper. You can either add the potato rounds and toss until they are coated with the butter mixture and then place them on the baking sheet in a single layer, with some space between the slices. Or, place the slices on the baking sheet and brush them on both sides with the garlic butter.
Bake for 25 to 30 minutes, until they are tender. Remove the potatoes and change the oven setting to the broiler. Use a potato masher, a fork, or the bottom of a glass to gently smush the potatoes so they end up being about ½-inch thick. The potatoes will lose their perfectly round shape as the potato mushes and the skin breaks a bit, but that’s ok.
Sprinkle the feta evenly over the potatoes. Broil them for 2 to 4 minutes, until the cheese gets melty and slightly browned. Serve hot or warm, with the red pepper flakes and parsley or scallions sprinkled over the top.