Judging by the symptoms, I’m guessing this is related to browsers using TLS false start, which is a client-side trick Google introduced to reduce the back-and-forth in TLS:
False Start is largely controlled by the browser and works by reducing the two round-trip passes of data described in official SSL specifications to a single round-trip pass. It did this by instructing the client to send Finished and first ApplicationData messages in a single dispatch rather than putting them in two distinct packages and sending the second only after getting confirmation from the server.
Google proposed False Start as an official standard to make SSL more palatable to websites that currently find it too expensive to offer. By abbreviating the handshake that negotiates the encryption key and other variables needed to protect data passing between the end user and website, False Start was intended to lower the performance penalty that many say comes from using the protocol.
From the relevant issue raised in Mozilla Firefox: (emphasis mine)
So far, an incomplete list of products that are known to have current or previous compatibility problems with False Start include (AFAICT): F5, A10, Microsoft TMG, Cisco ASA, ServerIron ADX, ESET, NetNanny, some configurations of Java’s SSL server implementation.