Choose a correct sized pan. See my article on How to Choose a Good Saute Pan.
If you have a pan that's big enough to hold all four fillets, that great. If not, cook two pieces at a time. You don't want to overcrowd the pan. This also can result in steaming rather than browning. On the other hand, if the pan is too big, the fillets may cook too quickly and burn.
Heat and Brown
Preheat your pan just a bit over low heat before you add the oil and butter. There are two reasons you do this.
All pans have hot spots. If you add butter or oil to a cold pan it can start burning before it melts or heats up. For example, if you add butter to the hot spot, it may burn before it completely melts and coats the bottom of the pan or if it is added to a non-hot spot and hits the hot spot it may burn.
There is an expression, "A watched pot never boils" which means if you stand there and watch a pot of water come to boil, it seems like it is taking forever. Our attention drifts and we get distracted. The same is true when heating up butter and oil in a pan. Have you ever added some cold butter to a cold pan, pushed it around a bit, became distracted and walked away only to have the butter burn? By preheating the pan you are ready to start cooking the moment you add your fat. Your attention is focused.
When the pan is hot, about 180 degrees F, add 1 tablespoon of oil and 1 tablespoon of butter. I like using a combination of oil & butter to get the extra flavor from the butter while the oil produces a nice crust and does not burn as quickly.
When the oil is hot but before the butter starts to brown, add the fillets skin side up and sear the skinless side until golden brown. This should take 3 - 4 minutes. With a wide spatula so not to break the fish, flip over to the other side and cook again for 3 - 4 minutes. Remove the fish fillets from pan and transfer to a plate.
When I prepared this at home the skin did not stick to the pan, but if it does, just scrape it off with your spatula and throw it out.
I am taking the luxury of assuming you have a pan big enough to cook all 4 pieces. If you don't and are cooking in batches, you are going to have to add another tablespoon of oil and one of butter. Don't worry about the extra fat, it adds to the flavor.
Now let's cook the fennel. Add the remaining tablespoon of olive oil to the pan and then the sliced fennel. Season with a little salt & pepper and cook 4 or 5 minutes until softened. Don't forget to give it a stir once in a while so it doesn't burn in the pan.
Add the minced garlic and cook for an additional minute. Remove from heat and add your wine. Not always necessary to remove the pan, but just to be on the safe side, I like to add wine to any hot dish off the source of heat.
Cover the pan with a tight fitting lid. If you don't have a matching lid for the pan, try some foil. Reduce the heat to medium and simmer the fennel for about 8 minutes until tender. The recipe suggests checking the fennel after 4 minutes to make sure it is not dry. If it is, just add a few tablespoons of water. When I made the dish it was not dry.
Add the tomatoes and the juice they were packed in, ½ cup of water, and 2 tablespoons of the cilantro. Replace the cover and simmer for an additional 5 minutes.
Remove the cover and taste for seasonings. Most often I find a dish needs a little salt to bring out the flavors. This is something you pick up with practice. You don't want to over salt a dish but if it tastes flat sometimes a pinch of salt will bring out the flavor.
Also remember if the food is hot, your taste buds won't register the flavor. Have you ever noticed that food right from the stove to the table doesn't always taste as good as after it has had time to cool down a bit. Some of it has to do with giving the flavors to meld together but it also has to do with how hot the food is and what our taste buds can register.
Return all the reserved sea bass to the pan skin side down even if you cooked them two at a time. At this point it doesn't matter. Add any of the accumulated juices that might be on the plate you held the fish. Don't throw out that additional flavor.
Once again cover the pan and cook at a low simmer until the sea bass is cooked through. About 4 -5 minutes.
Remove each piece carefully with your spatula and plate. Because of the incredible flavor from the fennel enhance broth, you may want to serve this dish in shallow soup bowls so you can soup out the broth at the end.
You could serve this with simple orzo pasta or even some perfect mashed potatoes. Add a couple of spoonfuls to the bowl and place the fish on top. Spoon the broth over the top and garnish with the remaining chopped cilantro and fennel tops. As I said at the beginning, this meal is one that you would be proud to serve anyone or be thrilled to be served in a nice restaurant.
"It's As Good As It Gets"