Combine the rice and 1 1/2 cups water in a medium saucepan; bring to a simmer over medium-high heat and cook until most of the water is absorbed, about 4 minutes. Reduce the heat to low; cover and continue cooking until all of the water is absorbed, about 12 more minutes. Uncover and transfer to a large bowl to cool slightly.
Preheat a large grill pan or grill over medium heat and brush with oil. Combine 2 tablespoons soy sauce, the vegetable oil, 1 1/2 teaspoons Sriracha and 1 teaspoon ginger in a bowl; rub all over the pork. Transfer the pork to the grill pan; cook about 8 minutes per side for medium doneness. Transfer to a cutting board and let rest 5 minutes; thinly slice.
Stir the remaining 2 tablespoons soy sauce, 1 1/2 teaspoons Sriracha and teaspoon ginger into the rice. Add the carrots, cucumber, lime juice and brown sugar and toss. Serve the rice and pork in the lettuce leaves; top with cilantro and drizzle with more Sriracha. Serve with lime wedges.