How would one transfer files larger than 2,147,483,646 bytes (~2 GiB) with Win32 TransmitFile()?

How would one transfer files larger than 2,147,483,646 bytes (~2 GiB) with
Win32 TransmitFile()?

Quoted from MSDN entry for TransmitFile:
The maximum number of bytes that can be transmitted using a single call to
the TransmitFile function is 2,147,483,646, the maximum value for a 32-bit
integer minus 1. The maximum number of bytes to send in a single call
includes any data sent before or after the file data pointed to by the
lpTransmitBuffers parameter plus the value specified in the
nNumberOfBytesToWrite parameter for the length of file data to send. If an
application needs to transmit a file larger than 2,147,483,646 bytes, then
multiple calls to the TransmitFile function can be used with each call
transferring no more than 2,147,483,646 bytes. Setting the
nNumberOfBytesToWrite parameter to zero for a file larger than
2,147,483,646 bytes will also fail since in this case the TransmitFile
function will use the size of the file as the value for the number of
bytes to transmit.
Alright. Sending a file of size 2*2,147,483,646 bytes (~ 4 GiB) with
TransmitFile would then have to be divided into two parts at minimum (e.g.
2 GiB + 2 GiB in two calls to TransmitFile). But how exactly would one go
about doing that, while preferably also keeping the underlying TCP
connection alive in between?
When the file is indeed <=2,147,483,646 bytes in size, one could just write:
HANDLE fh = CreateFile(filename, GENERIC_READ, FILE_SHARE_READ, NULL,
OPEN_EXISTING, FILE_ATTRIBUTE_NORMAL | FILE_FLAG_SEQUENTIAL_SCAN, NULL);
TransmitFile(SOCK_STREAM_socket, fh, 0, 0, NULL, NULL, TF_DISCONNECT);
to let Windows handle all the lower-level stuff (caching, chunking the
data up into pieces for efficient transmission etc. However, unlike the
comparable Linux sendfile() syscall, there is no immediately obvious
offset argument in the call (although the fifth argument, LPOVERLAPPED
lpOverlapped probably is exactly what I'm looking for). I suppose I could
hack something together, but I'm also looking for a graceful, good
practice Win32 solution from someone who actually knows about this stuff.
You can use the lpOverlapped parameter to specify a 64-bit offset within
the file at which to start the file data transfer by setting the Offset
and OffsetHigh member of the OVERLAPPED structure. If lpOverlapped is a
NULL pointer, the transmission of data always starts at the current byte
offset in the file.
So, for lack of a minimal example readily available on the net, which
calls are necessary to accomplish such a task?