Melt the butter in a small saucepan over medium heat until foaming. Whisk in the flour and cook for 2 to 3 minutes until it turns light tan and smells nutty. Stir in the curry powder, garam masala, and paprika, and cook for 1 to 2 minutes until fragrant and dark brown. Remove from heat and set aside.
Pat the chicken thighs dry with paper towels and season generously with salt and pepper. Heat 1 tablespoon of oil in a large Dutch oven over medium-high heat. Brown the chicken in a single layer for 3 to 4 minutes per side until golden. Transfer to a plate and set aside.
Reduce the heat to medium and add the remaining 1 tablespoon of oil to the pot. Add the sliced onion with a pinch of salt and cook for 10 to 12 minutes, stirring occasionally, until deeply golden and softened.
Add the minced garlic and grated ginger to the pot and stir for 60 seconds until fragrant. Add the grated apple with its juices and cook for 1 to 2 minutes, stirring occasionally, to soften slightly.
Return the browned chicken and any juices to the pot. Add the carrots, potatoes, chicken broth, soy sauce, and honey. Stir to combine and bring to a gentle boil over medium-high heat.
Reduce the heat to medium-low and simmer, covered with the lid slightly ajar, for 20 to 25 minutes, stirring every 5 minutes, until the vegetables are fork-tender and the chicken is cooked through.
Add the prepared roux to the pot one spoonful at a time, stirring after each addition until fully incorporated. Continue cooking over medium heat for 3 to 5 minutes until the sauce thickens to a glossy, gravy-like consistency. Taste and adjust seasoning with salt, soy sauce, or honey as needed.
Ladle the hot curry over steamed rice in deep bowls. Garnish with pickled red ginger on the side if desired. Serve immediately.