Season the chicken breasts evenly on both sides with the salt and pepper.

In a large 12 inch skillet, heat the vegetable or canola oil over medium heat until shimmering. Carefully add the seasoned chicken breasts to the hot skillet in a single layer. Cook until the chicken is fork tender and deeply golden brown on both sides, about 10 minutes, flipping once halfway through. Transfer the chicken to a warm serving platter, tent loosely with foil, and set aside.

Carefully spoon off and discard the remaining oil from the skillet.

With the skillet over low heat, squeeze the juice from the large lime directly into the pan. Cook gently, using a spatula to scrape up any browned bits from the bottom of the skillet, until the lime juice begins to bubble.

Using a wire whisk, add the cold unsalted butter to the skillet 1 tablespoon at a time, whisking constantly after each addition. Wait until each piece of butter is almost fully melted and incorporated before adding the next. Continue whisking until all the butter has been added and the sauce becomes opaque, thick, and creamy. Do not let the sauce boil.

Remove the skillet from the heat. Immediately stir in the finely chopped fresh dill and chives until combined.

Spoon the warm lime butter sauce generously over the reserved chicken breasts and serve immediately.