Combine the salt and water in a 4-quart container and stir until the salt has dissolved, 1 to 2 minutes. Add the trout filets, making sure they are submerged. Cover and refrigerate for 3 hours.
Remove the trout from the brine, rinse thoroughly, and pat dry. Place the trout, skin-side down, onto a wire rack set in a half-sheet pan. Dry in the refrigerator until the skin becomes shiny and somewhat tacky to the touch, 21 to 24 hours.
The next day, turn the smoker on so that it maintains a temperature of 150 to 160ºF. Place the trout on smoking racks, skin-side down, separating them by at least 1/4 inch and place into a smoker. Adjust heat as needed and cook until the fish is cooked through, has darkened in color, and has the desired level of smoke flavor, 2 1/2 to 3 hours.
Serve with a green salad, incorporate into a dip, or eat as an appetizer. Refrigerate unused portions in an airtight container for up to 1 week, or vacuum-seal and refrigerate for up to 1 month.