Pat chicken thighs dry; season with salt and pepper. Heat oil in a large Dutch oven or pot over medium-high until shimmering. Add chicken, skin-side down, and cook until golden brown, 6 minutes, flip and continue to cook until browned on second side, about 4 minutes. Transfer chicken to plate; set aside.
Reduce heat to medium, add onions and stir to coat; season with 1 teaspoon salt. Cook, covered, until softened and starting to brown, about 8 minutes, stirring occasionally. Stir in jalapeño, garlic and oregano; cover and cook until fragrant, 2 minutes, stirring halfway through.
Increase heat to medium-high. Add 4 cups water, scraping up any browned bits. Add tomatoes and their juices and browned chicken to pot. Bring to a boil over high then reduce to a simmer over medium; cook until chicken is cooked through, about 8 minutes.
Transfer chicken to a plate and let cool. Once chicken is cool enough to handle, remove and discard chicken skin and bones, shred meat and return to pot. Stir in cilantro and season soup with salt and pepper to taste.
Serve soup with avocado, crumbled Cotija and lime wedges.