Zest the two lemons and set aside in a small bowl. Remove the leaves from the stem of oregano (or parsley) and chop fine and set aside.
In a plate or dish, pour out ½ cup of flour (all purpose or self rising flour will work) and mix in 1 tsp salt and 1 tsp pepper.
Examine your chicken pieces, cut them or pound them down to about ½ inch thick. The very large breast pieces, it will be easier to just cut those in half (like a book) to make them thinner. Pat them dry with paper towels.
In a separate plate or in the tray that the chicken came in, generously salt and pepper both sides of the chicken.
Pour mixture of vegetable oil and butter together into a large skillet on the stove top and set heat to medium and allow to melt. When oil begins to sputter and bubble it is ready. You should be able to hear the chicken sizzle when you put the chicken in the skillet (see next step first though).
Coat each piece of chicken with a light coat of flour on each side and place into the frying pan. You may need to increase the pan heat to a little more than medium to keep the pan hot enough. Do not put more than 2 pieces into the pan at one time. Cook on each side for 2-3 minutes (until color develops) and then remove to a clean plate to rest. (If you are doing more than just 2 pieces of chicken for this recipe, then cook them two at a time until you have browned each on each side and set aside on the plate.).
Once each piece is removed, add 2 Tbs of fresh flour (do not use the flour you dredged the chicken in) to the pan and use a wooden spoon to stir it around and let it cook for a minute or so. (It will clump up in bits with any grease that is in your pan). Add the minced garlic to the pan as well and allow it to cook for a minute or so.
Add the broth all at once and whisk to incorporate it with the flour and garlic in the skillet. Once incorporated, whisk in the lemon juice that you can squeeze from the lemons that you zested in step 1 (use bottled lemon juice if you aren’t using fresh lemons). Be sure to scrape up any little bits of chicken that may be stuck to the bottom of the pan with your wooden spoon.
Once your sauce starts to bubble slightly and has cooked down to about ½ its original volume (this takes a few minutes), add the capers and the chicken pieces plus any juice that has collected in the plate as they rest. Spoon the sauce over the top of the chicken as the sauce starts to thicken and bubble. When your chicken reaches an internal temp of 160, Remove from the heat, tent over with aluminum foil and allow it to rest for about 3-5 minutes and serve hot and garnish with the lemon zest and oregano prepared in step one.