In a large skillet, heat the oil over medium heat. Add the shiitakes and cook, stirring, until tender, 4 to 5 minutes. Add the bok choy and 2 tablespoons soy sauce and cook, turning, until wilted, about 2 minutes. Transfer to a medium bowl.
Add the pork to the skillet and cook over medium-high heat, breaking up the meat, until no longer pink. Stir in the remaining 1 tablespoon soy sauce, then the reserved bok choy mixture; keep warm over low heat.
In a small bowl, combine the sour cream and chili sauce. In another skillet over medium-low heat, warm a tortilla just until hot, about 30 seconds per side. Transfer to a plate and slather with 2 tablespoons of the sour cream mixture. Spoon 1/2 cup rice down the center. Top with one-quarter of the bok choy mixture, drizzle 1 teaspoon sesame oil on top and then fold burrito-style. Repeat with the remaining tortillas and fillings. To serve, cut each burrito in half.