Bank a Die, Spin the Rest

Two fair six-sided dice are rolled to set their starting faces. The game then proceeds in turns: at the start of each turn you must bank at least one die that isn't yet banked (banking both at once is allowed), permanently locking its current face; then every unbanked die is rolled again. Play stops once both dice are banked, and your payoff is the sum of the two locked faces. What decision rule maximizes your expected payoff, and what is that maximum expectation?