Why did our Rebbe teach us to purposely eat 'gebrochtz' on the Last Day of Passover?!

Because the Rebbe wants us to taste Moshiach here and now!

More than this, it is permissible to eat "chametz" at the "Meal of Moshiach" which starts on Passover, if the meal extended, and one is now in his Passover meal, after dark! Amazingly, when this person who ate chametz after dark, during his final Passover meal, he will then reciter in his Grace After Meal, the Passover prayer, proclaiming it to still be, "On this day of Passover!"

And this is the taste of Moshiach, using who we are, including our ego, but to serve G-d!