Soak the rice in a bowl of hot water to cover for about 5 minutes. Drain in a sieve and rinse under cold water until the water runs clear; drain well.
In a blender or food processor, puree the chiles with the 1/2 cup water, cilantro, garlic, and salt until smooth. Pass the puree through a ﬁne sieve into a large liquid measuring cup; note the amount and set aside. Pour enough of the chicken broth into another liquid measuring cup to make 4 cups liquid total—you want to keep the two liquids separate, since you will add the puree ﬁrst.
Heat the oil in a medium saucepan over medium-high heat until hot but not smoking. Add the rice and cook, stirring often, until the color changes to a milky white, 3 to 4 minutes. Add the onion and sauté until slightly softened, 1 to 2 minutes.
Pour in the poblano puree and cook until it darkens, thickens, and has mostly been absorbed by the rice, 2 to 3 minutes. Stir in the chicken broth and lime juice, bring to a rolling boil, cover, and reduce the heat to the lowest possible setting. Cook until most of the liquid has been absorbed but there is still some moisture in the pan, about 15 minutes. The rice should be cooked and tender; if it is not but all the liquid has been absorbed, add 2 tablespoons water, cover again, and cook for a couple more minutes. Let the rice rest, covered, for at least 5 minutes.
Fluff the rice with a fork and serve with the crumbled cheese on top, if using.