Lime Dill Butter Sauce on Tilapia or Chicken

My husband got some pots of fresh herbs for the backyard this year (Yes, HE is the one who cares about such things!) and the dill is flourishing a bit too much, so this has become a regular, delicious staple in our home.  It is so quick and fresh tasting!  We have been having it on fish lately because it is lower fat than chicken, but it is amazing on chicken as well.  It has become one of our family's staples.  I originally got the recipe from a friend, Sarah Jacobs Huff.  We "eye" EVERYTHING and do approximate amounts on this recipe.  It's not an exact science.

Lime Dill Butter Sauce for Chicken or Fish

4 boneless skinless chicken breasts
~OR~
4 tilapia fillets (We use the frozen ones from Sam's Club.  Good and WAY cheaper than fresh fish!)

Salt
Pepper
Vegetable Oil (for cooking)
1/4 cup butter (1/2 stick)
1 Tablespoon lime (or lemon) juice
1/2 teaspoon dill weed, chopped up

Dry off the meat with a paper towel.  (This is an essential step in "browning!)  Salt and pepper the meat.  Use a bit of vegetable oil to brown the meat in a skillet.  Remove browned meat and keep warm.  Add butter and lime juice (or lemon juice if you're in a pinch) to the skillet; cook and stir until butter is melted.  It tastes even better if you brown the BUTTER a bit.  Cook until just brown but not burned.  Stir in the dill weed and drizzle over the meat.  YUM!!!