refactor(mcp): improve StdioServerTransport shutdown and error handling

- Add specific error categorization for pipe closed errors
- Enhance shutdown sequence with proper stream and scheduler cleanup
- Add debug logging for shutdown-related stream errors
- Improve isClosing state management

mcp/src/main/java/org/springframework/ai/mcp/server/transport/StdioServerTransport.java

@@ -171,9 +171,18 @@ public class StdioServerTransport implements McpTransport {
 				}
 			}
 			catch (IOException e) {
-				if (!isClosing) {
+				// Check isClosing before the error occurs to properly categorize it
+				boolean wasClosing = isClosing;
+				isClosing = true;
+				if (!wasClosing && e.getMessage().equals("Pipe closed")) {
+					logger.debug("Stream closed during shutdown", e);
+				}
+				else if (!wasClosing) {
 					logger.error("Error reading from stdin", e);
 				}
+				else {
+					logger.debug("Stream error during shutdown", e);
+				}
 			}
 			finally {
 				isClosing = true;
@@ -209,6 +218,9 @@ public class StdioServerTransport implements McpTransport {
 							logger.error("Error writing message", e);
 							sink.error(new RuntimeException(e));
 						}
+						else {
+							logger.debug("Stream closed during shutdown", e);
+						}
 					}
 				}
 				else if (isClosing) {
@@ -236,16 +248,14 @@ public class StdioServerTransport implements McpTransport {
 		return Mono.fromRunnable(() -> {
 			isClosing = true;
 			logger.debug("Initiating graceful shutdown");
-		})
+		}).then(Mono.defer(() -> {
-			// .then(Mono.defer(() -> {
+			// First complete the sinks to stop processing
-			// inboundSink.tryEmitComplete();
+			inboundSink.tryEmitComplete();
-			// outboundSink.tryEmitComplete();
+			outboundSink.tryEmitComplete();
-			// return Mono.delay(Duration.ofMillis(100));
+			return Mono.delay(Duration.ofMillis(100));
-			// }))
+		})).then(Mono.fromRunnable(() -> {
-
-			.then(Mono.fromRunnable(() -> {
 			try {
-
+				// Dispose schedulers first
 				inboundScheduler.dispose();
 				outboundScheduler.dispose();
 
@@ -257,14 +267,27 @@ public class StdioServerTransport implements McpTransport {
 					outboundScheduler.disposeGracefully().block(Duration.ofSeconds(5));
 				}
 
+				// Only after schedulers are disposed, close the streams
+				try {
+					if (inputStream != System.in) {
+						inputStream.close();
+					}
+					if (outputStream != System.out) {
+						outputStream.flush();
+						outputStream.close();
+					}
+				}
+				catch (IOException e) {
+					// Log but don't throw since we're shutting down
+					logger.debug("Error closing streams during shutdown", e);
+				}
+
 				logger.info("Graceful shutdown completed");
 			}
 			catch (Exception e) {
 				logger.error("Error during graceful shutdown", e);
 			}
-			}))
+		})).then().subscribeOn(Schedulers.boundedElastic());
-			.then()
-			.subscribeOn(Schedulers.boundedElastic());
 	}
 
 	@Override