To prepare the brine, add salt to ice water and stir until the salt is dissolved. Pour the water into a Ziploc bag, add olive oil and stir. You can add herbs and spices to the brine as well to add more flavor. Dill and black pepper are commonly added.
Add salmon fillets, push out as much air as possible and seal the bag. Refrigerate for about 30 minutes.
Remove salmon fillets from the Ziploc bag, pat dry with a paper towel and transfer into another bag. Add 1 tablespoon of olive oil to avoid the fillets sticking to each other. Gently remove as much air from the bag as possible and seal. You want to be careful not to squeeze the fillets.
Immerse the sealed bag into a pot of preheated water and cook at 122F (or higher depending on the level of doneness you want) for one hour, frequently stirring to avoid cold and hot spots, and adjusting the temperature as needed (see instructions above). Alternatively, use an immersion circulator which will make things much easier.
Shortly before the cooking is done, preheat a large skillet with 1 tablespoons of olive oil.
Remove the salmon fillets from the bag and sear skin side down over high heat for 45 seconds.
Serve immediately.