Heat 1 tablespoon of the oil in the largest skillet you have (at least 12 inches) over medium-high heat until shimmering.
Add the onions and season with ¼ teaspoon of the salt.
Cook, tossing often with tongs to prevent burning, until they begin to soften and brown, about 3 minutes.
Add the peppers and garlic until fragrant. Season with ¼ teaspoon of the salt.
Transfer the vegetables to a plate.
Return the pan to medium-high heat (no need to clean) and add the remaining 1 tablespoon oil. Add the chicken and season with the remaining ½ teaspoon of salt, oregano, and cumin.
Cook undisturbed until the chicken has some golden-brown color, flipping halfway through, about 6 minutes total.
Return the vegetables to the pan and add the lime juice.
Cook until the chicken is cooked through and the peppers are soft, about 2 minutes more.
Serve the fajitas. Serve the chicken and vegetables in warmed tortillas.
