Step 1: Preheat your oven to 400°F (200°C). Line a baking sheet with parchment paper.
Step 2: On a lightly floured surface, gently unroll the thawed puff pastry sheet. If needed, lightly roll it with a rolling pin to a slightly larger rectangle (about 10x12 inches).
Step 3: In a small bowl, whisk together the grated Parmesan cheese, garlic powder, onion powder, black pepper, and salt.
Step 4: Brush the puff pastry generously with the beaten egg, covering the entire surface.
Step 5: Evenly sprinkle the cheese and spice mixture over the egg-washed pastry. Gently press the cheese mixture into the pastry with your hands to help it adhere.
Step 6: Using a sharp knife or a pizza cutter, cut the pastry lengthwise into 1/2-inch wide strips.
Step 7: Carefully twist each strip several times and place them about 1 inch apart on the prepared baking sheet.
Step 8: Bake for 12-15 minutes, or until golden brown and puffed. Let cool slightly on the baking sheet before transferring to a wire rack to cool completely.
Step 9: While cheese straws are baking, preheat a separate oven or use the same one at 300°F (150°C). Line a baking sheet with parchment paper.
Step 10: In a large bowl, toss the mixed nuts with olive oil until evenly coated.
Step 11: In a separate small bowl, combine the brown sugar, smoked paprika, ground cumin, cayenne pepper (if using), and sea salt.
Step 12: Sprinkle the spice mixture over the nuts and toss well to ensure every nut is coated.
Step 13: Spread the seasoned nuts in a single layer on the prepared baking sheet.
Step 14: Roast for 15-20 minutes, stirring halfway through, until the nuts are fragrant and lightly toasted. Keep a close eye on them to prevent burning.
Step 15: Remove from the oven and let cool completely on the baking sheet. The spices will crisp up as they cool. Serve both appetizers together for a fantastic New Year's spread!