Place chicken wings on a 16-inch square of cheesecloth; tie opposite corners of cloth together to form a bundle.
Heat a 6- to 8-quart stovetop pressure cooker over medium-high, or set an electric pressure cooker to sauté. Add bacon and cook until some fat is rendered, about 6 minutes. Add lemongrass, cilantro stems, ginger, and garlic, and cook, stirring occasionally, until golden and fragrant, about 3 minutes. Stir in fish sauce, scraping up any browned bits with a wooden spoon. Stir in the water, chicken wing bundle, chicken thighs, mushrooms, and salt.
Secure lid. Manually set cooker to 25 minutes and let it come to pressure. Once time is complete, turn off, quickly release pressure, then remove lid.
Let chicken stand in liquid for 10 minutes. Remove chicken wing bundle. Skim any fat from surface of soup, if desired.
Transfer chicken thighs to a cutting board. Remove skin from chicken thighs and pull meat from bones. Using two forks, shred chicken into pieces (discard bones and skin). Return chicken to soup. Stir in lime juice, top with mint leaves and avocado, and serve with lime wedges.