Place the ingredients in a medium saucepan over medium heat. When the water comes to a boil, cover the pan and let the bulgur cook on low heat for around 15 minutes. Bulgur cooked this way has a softer, fluffier consistency. If you'd like it to be chewier, turn off the heat right after the water comes to a boil and let the bulgur soak for about 15 minutes.
Place the basil leaves and tomato in a food processor and with the motor running, drizzle in the olive oil. The pistou will have a pretty runny consistency.
To put together the pilaf, fluff the bulgur with a fork and then gently stir in as much pistou as you need to get your flavour kick.