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We consider a source node broadcasting to two receivers over a general erasure channel with receiver feedback. We characterize the capacity region of the channel and construct algorithms based on linear network coding (either randomized or depending on channel dynamics) that achieve this capacity. We then consider stochastic arrivals at the source for the two destinations and characterize the stable throughput region achieved by adapting the same algorithms that achieve capacity. Next, we modify these algorithms to improve their delay performance and characterize their stable throughput regions. Although the capacity and stability regions obtained by the algorithms are not always identical (because of the extra overhead needed for the algorithms to handle stochastic traffic), they are within a few bits of each other and have similar forms. This example exhibits an unusual relationship between capacity and stability regions and extends similar prior studies for multiple access channels.