When I'm riding in the winter I'm usually riding either a MTB or hybrid with flat bars and barends so I can't comment on shifting and braking in mittens on drop bars. Braking is unaffected, and any shifting done with the thumb is fine. It can be difficult to fit the mittened fingers between the brake lever and bar to get to the other shifters so I usually let go with the opposite hand, reach under the bar and use the other thumb to do those shifts. Of course if it is slushy it is all irrelevant because the ice build up means you can't shift anyway. Who said you had to spend money to get a single speed bike?