Rounded off a caliper bracket bolt: best extraction method?

Stop before it gets worse. Every turn with a 12-point socket is rounding it further, so put that socket away and switch to a six-point bolt extractor socket (the spiral "bolt-out" type), one size that you have to tap on. Wire-brush the rust off the bolt head first so the extractor can actually bite, then soak it with penetrating oil. Give it at least 15-20 minutes and a second application, tapping the bolt head lightly helps the oil creep in.

Hammer the extractor on until it is fully seated on the head. Then use a breaker bar with slow, steady pressure. A short burst from an impact wrench on a low setting can shock it loose, but go easy, snapping the bolt off in the bracket turns a 20-minute job into a drilling job.

Heat is the last resort. A careful, localized heat on the bracket around the bolt can break the rust bond, but keep the flame away from the rubber caliper boots, the brake hose, and the ABS sensor wiring. If the bolt still will not move, that is the point to hand it to a shop with a torch and a lift rather than break something expensive.

Once it is out, do not reuse it. Fit a new caliper bracket bolt and torque it to the spec in the model-specific service information for your truck, clean the bracket and slide surfaces while you are in there.

Full walkthrough for the brake job on your generation of F-150 here: https://ezcarfix.com/guides/ford/f-150/2009-2014/front-brake-replacement-complete