Preheat the broiler. Line a baking sheet with foil. Put the potatoes in a medium saucepan and cover with cold water by 1 inch; season with salt. Bring to a boil, then reduce the heat and simmer until the potatoes are tender, 10 to 12 minutes, adding the peas in the last 1 minute of cooking. Drain the potatoes and peas; reserve the saucepan.
Meanwhile, toss the red onion, vinegar, 2 tablespoons water and a big pinch of salt in a bowl; set aside. Mix the turkey with 1/4 cup tikka masala sauce, 1/2 teaspoon salt, 2 tablespoons cilantro and 1/4 teaspoon cumin. Shape into four 1/4-inch-thick patties. Put on the prepared baking sheet; spread with the remaining 1/4 cup tikka masala sauce. Broil until cooked through, 6 to 8 minutes.
Melt the butter in the reserved saucepan over medium-high heat. Stir in the remaining 1/2 teaspoon cumin, then add the potatoes and peas. Cook, stirring, until hot, 2 to 3 minutes. Stir in the remaining 1/3 cup cilantro; season with salt.
Drain the red onion. Serve the burgers on the buns with the mayonnaise and red onion. Serve with the potatoes and peas.