Bring the coconut milk, 3/4 cup water and the lemongrass or ginger to a boil in a medium saucepan. Add the rice and return to a boil. Stir, then cover and reduce the heat to low. Simmer until the liquid is absorbed and the rice is tender, about 15 minutes. Remove from the heat and let sit, covered, 10 minutes. Fluff with a fork.
Meanwhile, whisk 1/4 cup soy sauce, the lime juice and sugar in a small bowl. Toss the cucumbers with 1 tablespoon basil and the remaining 1 teaspoon soy sauce in a medium bowl. Set aside.
Heat 1 tablespoon vegetable oil in a medium skillet over high heat.
Add the beef and cook, breaking up the meat, until browned and cooked through, about 5 minutes. Transfer to a bowl; set aside.
Reduce the heat to medium high. Add the remaining 1 tablespoon vegetable oil to the skillet. Add the bell pepper and cook, stirring occasionally, until golden, about 5 minutes. Add the garlic and cook until golden, 2 minutes. Increase the heat to high and return the beef and any accumulated juices to the skillet. Stir in the soy sauce mixture and cook until the liquid evaporates, 2 minutes. Add the remaining basil and cook, stirring, until just wilted, about 1 minute. Serve the beef and cucumber salad over the rice.