Pat the drumsticks completely dry with paper towels to ensure crispy skin. Rub each piece evenly with olive oil, coating every surface. In a small bowl, combine paprika, garlic powder, onion powder, dried thyme, salt, and black pepper. Generously coat each drumstick with the spice blend, pressing gently so it adheres. Let sit for a few minutes while preparing the grill.

Heat the grill to medium-high heat and lightly oil the grates. Place drumsticks skin-side down and cook undisturbed for about 8 minutes to achieve caramelization. Flip and cook for another 20 to 22 minutes, turning occasionally. The internal temperature should reach 165°F when checked with a meat thermometer. The skin should be dark, blistered, and crispy.

While the chicken cooks, place peeled potatoes in a large pot and cover with cold salted water. Bring to a boil and cook for 15 to 20 minutes until completely tender and easily pierced with a fork. Drain well and return to the pot. Add butter and milk, then mash until creamy with no large lumps. Season with salt and pepper to taste.

In a medium bowl, whisk together mayonnaise and plain yogurt until well combined. This creates a lighter, tangier dressing than mayo alone. Season with salt and black pepper. Stir well. The dressing is best made right before serving.

Add diced cucumber and halved cherry tomatoes to the bowl with the creamy dressing. Gently toss everything until the vegetables are evenly coated. Taste and adjust seasoning as needed.

Place a generous spoonful of mashed potatoes on each plate. Nestle the grilled drumsticks next to the potatoes and top with a pinch of fresh black pepper and a drizzle of olive oil. Spoon the creamy salad mixture onto the side of each plate. Serve immediately while the chicken is still warm.