[Feature] Flush the speech WebSocket on input.done without closing it (#5517)

Signed-off-by: iancarrasco-b10 <ian.carrasco@baseten.co>

docs/user_guide/examples/online_serving/text_to_speech.md

@@ -422,12 +422,19 @@ curl -X POST http://localhost:8091/v1/audio/speech \
 Raw PCM streaming requires `stream_format="audio"`, `response_format="pcm"`, and `async_chunk: true` on the stage config (default in `qwen3_tts.yaml`). `speed` is not supported when streaming.
 
 ### Streaming WebSocket
-The `/v1/audio/speech/stream` endpoint accepts text incrementally, splits it at sentence boundaries, and emits one PCM stream per sentence:
+The `/v1/audio/speech/stream` endpoint accepts text incrementally and synthesizes the buffered text as one continuous request on `input.done`:
 ```bash
 python qwen3_tts/streaming_speech_client.py --text "Hello world. How are you? I am fine."
 python qwen3_tts/streaming_speech_client.py --text "..." --simulate-stt --stt-delay 0.1
 ```
 
+`input.done` flushes without closing, so repeating `--text` synthesizes several utterances over one connection:
+```bash
+python qwen3_tts/streaming_speech_client.py \
+    --text "First utterance." \
+    --text "Second utterance, same connection."
+```
+
 ### Gradio demos
 ```bash
 ./qwen3_tts/run_gradio_demo.sh                              # CustomVoice (default)