Preheat the oven to 350 degrees F. Combine the cucumber, 1/4 teaspoon garam masala, 1/2 teaspoon salt and a few grinds of pepper in a medium bowl; toss to coat. Stir in the yogurt and set aside. Combine the chutney, chopped jalapeno, cilantro and lime juice in a separate bowl; stir to combine.
Season the chicken on both sides with the remaining 1 3/4 teaspoons garam masala and a generous pinch each of salt and pepper. Heat a large skillet over medium-high heat, then add the olive oil. Add the chicken and cook until browned, 4 to 5 minutes. Flip and cook until browned on the other side, 2 to 3 more minutes. Reduce the heat to medium and add 1/4 cup water. Cover and cook until the chicken is cooked through, 10 to 12 minutes. Transfer the chicken to a cutting board and let rest 5 minutes.
Meanwhile, warm the naan in the oven on a baking sheet. Slice the chicken and divide among the naan. Top with the chutney mixture, red onion, sliced jalapeno and more cilantro. Serve with the cucumber mixture.