Set up your Big Green Egg for raised direct cooking. Preheat the dome temperature to 400°F.
Pat the chicken thighs dry with paper towels. Season both sides with kosher salt and black pepper.
Add the hot sauce and butter to a small saucepan. Simmer over medium heat until reduced by about half.
Remove from heat and set aside.
Bring the chicken stock, butter, salt, and pepper to a boil. Stir in the couscous.
Remove from heat, cover, and let sit until tender.
Spread the couscous out and cool to room temperature.
Transfer cooled couscous to a bowl.
Add the egg, chopped green onions, and Parmesan cheese.
Mix until fully combined. Form into 4 evenly sized patties.
Place the chicken thighs skin side down on the Big Green Egg.
Cook until the fat renders and the skin becomes crispy. Flip the chicken.
Begin brushing with the hot sauce butter glaze. Continue cooking and glazing in layers.
Cook until the internal temperature reaches 175°F–185°F. Remove and rest.
Heat a cast iron skillet on the Big Green Egg. Add olive oil.
Cook the couscous cakes until golden brown and crispy. Flip and repeat.
Whisk together olive oil, Dijon mustard, red wine vinegar, honey, salt, and pepper.
Toss the arugula with the vinaigrette. Add the blue cheese.
Place the couscous cakes on serving plates.
Top with the dressed arugula salad. Add thinly sliced pear.
Top with the glazed chicken thighs. Serve immediately.