If you get an A on both then you already passed for DAS on the first hand. But yes, without any heat considerations you would obviously double on the 11 as long as the count called for it. It would need to be pretty high with EV indices and monster high for RA indices. I personally would never make this play for longevity concerns.
You have to check the house rules to see if doubling on a soft 21 is allowed. Several of my usual haunts specifically prohibit such a play. If in doubt, just ask the dealer or pit boss... they'll never steer you wrong ;-)
I have only seen doubling on a blackjack prohibited. In some places they say doubling on a natural is prohibited. I wouldn't think this is the same thing but perhaps the pit interprets it as the same. I believe we frequent the same casinos so perhaps I am mistaken.
I pulled this off a Bally's a few years back and judging from the pit reaction it wasn't a good idea. I was young, niave and slightly capricious then. It's bad enough chopping the tens but then doubling on "21". Only if you plan on that being your last hand and not playing in that venue for sometime.