You have the perfect controller for diagnosing this, because the paddles are removable and that gives you a test nobody with fixed rear buttons can run.
The paddles on an Elite Series 2 are magnetically attached levers. Each one presses a switch on the controller body. So there are two possible culprits: the paddle lever itself, which can be slightly bent or worn at its contact point so it bounces against the switch, or the switch underneath, which can be suffering contact bounce in exactly the way a worn face button does. Both produce two events from one press and they are indistinguishable from the symptom alone.
The diagnostic is to move the paddle. Take the top left paddle off and fit it in the top right position, and move the top right paddle to top left. Then re-run the test. If the double fire moves with the paddle to the right side, the paddle is the faulty part. If the double fire stays on the top left position with a known-good paddle fitted, the switch underneath is the fault. That single swap costs nothing and tells you exactly which part to buy, which is far better than ordering both and hoping.
If it is the paddle, replacement paddle sets are available and inexpensive, and you can also inspect the lever where it meets the switch for a worn or deformed contact face.
If it is the switch, that is inside the controller body and a more involved repair, but worth knowing you are also inside a longer warranty than most pads: Microsoft extended the warranty on this model specifically because of known issues with it. So check your purchase date before you pick up a screwdriver, because a warranty repair on an Elite is worth considerably more than on a standard pad.
One thing to rule out first, because it takes thirty seconds: check your active profile has not got a turbo or rapid-fire setting applied to that button, which would produce the same double input from a perfectly healthy paddle.
