Pat the chicken thighs dry with paper towels: Season both sides with 3/4 teaspoon of the salt, the black pepper, cumin, and oregano.
Heat the olive oil in a 12-inch deep skillet over medium-high heat until shimmering.
Add the chicken thighs in a single layer: Cook for 4 to 5 minutes without moving them excessively, until the underside is deeply golden.
Flip and cook for another 3 minutes. The chicken does not need to be fully cooked at this stage. Transfer it to a clean plate.
Reduce the heat to medium: Add the sliced onion to the same skillet and cook for 4 to 5 minutes, stirring occasionally, until softened and lightly golden.
Stir in the minced garlic and cook for about 30 seconds, until fragrant but not browned.
Pour in the salsa verde and chicken broth: Stir well, scraping the bottom of the skillet to loosen the browned bits.
Nestle the chicken thighs into the sauce along with any juices from the plate. The sauce should come about halfway up the sides of the chicken.
Bring the sauce to a gentle simmer, then reduce the heat to medium-low. Cover and cook for 10 minutes.
Remove the lid and continue simmering for 5 to 8 minutes, until the sauce has thickened slightly and the chicken is fully cooked.
Insert an instant-read thermometer into the thickest part of a chicken thigh. The chicken must reach at least 165°F (74°C).
If the salsa verde is still thinner than desired, transfer the cooked chicken to a clean plate and simmer the sauce uncovered for another 2 to 4 minutes. If it becomes too thick, add a splash of chicken broth.
Remove the skillet from the heat: Stir the lime juice and chopped cilantro into the sauce. Taste and add the remaining 1/4 teaspoon salt only if needed.
Return the chicken to the sauce if it was removed and let it rest for 5 minutes. Garnish with avocado, cotija, extra cilantro, lime wedges, or jalapeño as desired.