Place potato and butternut squash in saucepan and cover with 1 inch of water. Bring to boil and cook until tender, about 10 minutes. Drain and transfer to a heatproof bowl and mash. Stir in cheese, flour, garlic, herbs, salt and pepper.
Spread breadcrumbs evenly in a shallow dish or baking sheet. Form mixture into patties, about 3-inches across and scant 1/2-inch thick. Freeze 15 minutes. Using a sharp knife, cut Jack O’Lantern faces in each patty. Place in breadcrumbs and turn to coat both side.
Heat 2 tablespoons olive oil in a nonstick skillet over medium. Cook patties, 2 to 3 at a time for two minutes per side until golden brown, flipping gently with an offset spatula. Replace oil between batches to prevent scorching. Serve immediately.