Mix 1/4 cup yogurt, 1 tablespoon olive oil and the tandoori paste in a large resealable plastic bag. Add the turkey and shake to coat. Set aside to marinate, 20 minutes.
Meanwhile, pulse the ginger and garlic in a mini food processor to make a paste. Heat a large saucepan over medium heat. Add 1 tablespoon olive oil, then the ginger paste. Cook until the paste sizzles, 1 minute. Add the tomato paste and garam masala and stir until sizzling, 1 minute. Stir in the lentils. Add 1 cup water and season with 1/2 teaspoon salt. Bring to a simmer and cook until thickened, about 7 minutes. Stir in the kale and 3/4 cup cilantro; cook until the kale wilts, about 2 minutes. Keep warm over low heat.
Heat a large skillet over medium-high heat. Add the remaining 1 tablespoon olive oil. Remove the turkey from the marinade, letting the excess drip off, and add to the skillet. Cook, turning once, until just cooked through, 4 to 5 minutes. Serve over the lentils and top with the remaining 1/4 cup each yogurt and cilantro.