In a 3- to 4-quart pan over high heat, bring 3 1/2 cups water, the rice, and 1/2 teaspoon salt to a boil. Cover, reduce heat to low, and simmer until liquid is absorbed and rice is tender to bite, about 45 minutes.
Meanwhile, rinse chicken and pat dry; cut into 1-inch chunks. Trim and discard root ends, tough green tops, and tough outer layers from leeks. Split leeks in half lengthwise and rinse well under running water, flipping layers to release grit; thinly slice crosswise.
Pour oil into a 10- to 12-inch frying pan over medium-high heat. When hot, add chicken and stir until no longer pink in the center (cut to test), about 7 minutes. With a slotted spoon, transfer chicken to a rimmed plate or bowl. Reduce heat to medium and add leeks and garlic to pan; stir until leeks are limp, about 5 minutes. Add mushrooms and stir until mushrooms are browned and have released their juices, about 5 minutes. Add wine and bring to a simmer, scraping bottom of pan to release any browned bits.
Return chicken to pan. Add cream and soy sauce and simmer, stirring occasionally, until liquid is slightly thickened, 2 to 3 minutes. Add salt and pepper to taste.
Spoon rice into four wide, shallow bowls and top equally with chicken, vegetables, mushrooms, and sauce.