Spray 5- to 6-quart slow cooker with cooking spray. In 10-inch skillet, heat vegetable oil and sesame oil over mediumhigh heat. Add garlic, gingerroot and shiitake mushrooms; cook 4 to 5 minutes, stirring frequently, until mushrooms begin to brown. Spoon mixture into slow cooker. Add chicken to skillet; cook 6 minutes, turning once, until lightly browned. Place chicken on mushroom mixture in slow cooker.
Add lemongrass paste, canned mushrooms and broth to slow cooker; stir to combine.
Cover; cook on Low heat setting 4 hours.
Remove chicken from slow cooker to plate; shred chicken, using 2 forks. Add shredded chicken and onions to slow cooker; stir to combine.