I start braking first, and do downshifts during the braking period (but bear in mind that the time between starting braking, and starting downshifting, could be a fraction of a second). Starting the braking first is in the interest of not over-revving the engine following the downshift. But ... Having watched (and cringed) during track days, perhaps it is necessary to point out a couple of things that might not be otherwise obvious.
If you are braking HARD enough, then the time that it takes to complete the downshifts is going to (almost) automatically keep the engine in the right RPM range without over-revving, because you are knocking off enough speed during each downshift to be in the appropriate speed range for the next lower gear anyhow, as you complete each downshift. Read Amazon's post again. She may be starting the downshifting action together with starting the braking action ... but she is braking hard, and by the time the downshift is completed, speed has been knocked off. I've heard her ride. She's not killing her engine (any more than roadracing inherently does, LOL)
Also, if you are braking HARD enough, there is not much weight on the rear wheel. My bike does not have a slipper clutch. It does, however, slide the rear tire under braking if I get a downshift a bit wrong.
From watching track days ... If you are rolling off the throttle 200 metres too early for the corner and then downshifting (and spiking engine revs way into redline, making me cringe) and only then braking gently ... you are doing it wrong. Brake later. Brake harder. Do the downshifts together with braking, spaced out during the braking period, not before you even start applying brakes the way I've seen some track-day riders do.
Remember, a slipper clutch only cushions downshifts - it does not prevent downshift-overrevving from mechanically back-driving the engine because you downshifted into an inappropriate gear for your current road speed.