What is the method for making up two consecutive Ramadan fasts, given the obligation to fast sixty consecutive days for each day, and is this make-up invalidated by interrupting the fasting?

If you broke your fast by eating and drinking, then committed indecency, then according to the Shafi'is, you are only required to make up for the missed fasts (qada'), and no expiation (kaffara) is due from you. This is permissible out of necessity. If you offered a single expiation for all those days, that would be good. You are obligated to make up for those days immediately.

However, if the breaking of the fast was due to sexual intercourse, then according to some scholars, offering a single expiation would suffice for you. The majority, however, require an expiation for each day (fasting two consecutive months, or feeding sixty poor persons if you are unable to fast). The expiation is due immediately.

And if, according to the Shafi'is, only making up the fasts (qada') is required of you, then according to the majority of scholars, in addition to making up the fasts, you are also required to feed one poor person for each day you delayed making up the fast.