c++: BOOST_ASIO Server reply isn’t as expected?

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I think handle_read() calls handle_write() recursively?

is close but not quite correct as there is no recursion here. The documentation explains this nicely:

Regardless of whether the asynchronous operation completes immediately
or not, the handler will not be invoked from within this function.
Invocation of the handler will be performed in a manner equivalent to
using boost::asio::io_service::post().

Added emphasis is mine. Instead of recursion, It is better to think of these concepts as chaining since one operation such as async_write() is initiated in the handler of another, such as async_read(). The exact specifics depend on the protocol in use.

If you want the server to send the string 0123456789 to the client, fill your buffer before invoking async_write().