Pat the chicken dry and season with salt and pepper on all sides.
Melt the butter in a small skillet and stir in 1 tablespoon tikka masala spice and the tomato paste. Cook, stirring, until fragrant, about 1 minute. Remove from the heat. Stir the ginger, garlic, lemon juice and yogurt together in a large bowl. Slowly whisk in the butter mixture. Add the chicken and sprinkle with more salt and pepper. Toss to coat. Marinate the chicken for 4 to 6 hours.
When you're ready to fry, remove the chicken from the refrigerator and let set at room temperature while you prepare the remaining ingredients.
In a small saucepan, whisk together 2 cups vegetable oil, the remaining 3 tablespoons tikka masala spice, 6 tablespoons of the chili powder and the brown sugar. Place over low heat until ready to use.
In a medium saucepan, whisk together the vodka sauce and 1/4 cup of the chili oil. Bring to a simmer, then keep over low heat until ready to serve. Season with salt and pepper to taste.
In a shallow dish, combine the flour, baking powder, 2 tablespoons salt, onion powder, sweet paprika, remaining 1 tablespoon Indian chili powder and black pepper to taste.
Heat a few inches vegetable oil in a large Dutch oven until it registers 350 degrees F on a deep-fry thermometer.
Remove the chicken from the marinade and shake off any excess marinade. Coat the chicken pieces one at a time in the flour mixture and place on a baking sheet. Set a wire rack over another baking sheet for draining the chicken after it fries. Carefully lower the coated chicken into the oil. The temperature will drop to between 300 and 325 degrees F; adjust the heat to maintain this temperature. Fry the chicken without turning for 2 to 3 minutes to set the crust. Continue to cook, carefully turning the pieces as needed with tongs, until golden brown and cooked through, 12 to 14 more minutes. An instant-read thermometer inserted into the thickest part of each piece should register 165 degrees F. Transfer the fried chicken to the wire rack to drain. Sprinkle with salt.
To serve, place a piece of bread on a plate, top with a large spoonful of vodka tikka masala sauce and place 2 pieces of chicken on the bread. Drizzle with the chili oil.