When crisping potatoes, it's important not to crowd the pan, so you may need to cook them in two batches. If so, for the second batch, return the heat to high and add more oil or butter if necessary.
Place potatoes in a large pot; cover with water. Add enough salt so that the water tastes like the ocean, about 2 tablespoons. Cover, and bring to a boil.
Remove from heat. Add garlic scapes. Let sit 10 minutes; the potatoes should be tender when pierced with a knife. Drain, and separate potatoes from scapes. When potatoes are cool enough to handle, cut in half.
Heat oil in a skillet over high heat. Add potatoes, in batches if necessary, cut sides down, and cook until they begin to brown, about 1 minute. Reduce heat to medium, and add 1 tablespoon butter. Cook until potatoes are golden and crisp, stirring occasionally, 7 to 8 minutes.
Add garlic scapes and 1 tablespoon butter to pan; raise heat to make potatoes extra crisp. Season with salt and pepper.