Pat the drumsticks dry with paper towels. Combine paprika, garlic powder, onion powder, salt, black pepper, and chili powder in a bowl. Mix in olive oil and lemon juice to create a paste. Rub the mixture all over each drumstick, coating evenly and getting some seasoning under the skin. Let sit for at least 10 minutes, preferably 30 minutes.

Cut potatoes into thick fries. Soak them in cold water for about 5 minutes to remove excess starch, then pat completely dry. Toss with olive oil and salt right before cooking.

Arrange lettuce on a plate. Add cucumber slices, tomato wedges, and red onion rings. Keep refrigerated until ready to serve.

Whisk together mayonnaise, ketchup, and hot sauce in a small bowl until smooth and well combined. Prepare ahead or right before serving.

Preheat grill to medium-high heat, about 400 degrees Fahrenheit (or 425 degrees in oven). Place drumsticks on grill and cook for 25-30 minutes, turning every 5-7 minutes until internal temperature reaches 165 degrees Fahrenheit. Cook fries on a grill pan or directly on the grates for about 20-25 minutes, stirring occasionally until golden and crispy. Both should finish cooking around the same time.