Use shared is_async_callable instead of inspect.iscoroutinefunction (#2389)

src/mcp/shared/_callable_inspection.py

@@ -0,0 +1,33 @@
+"""Callable inspection utilities.
+
+Adapted from Starlette's `is_async_callable` implementation.
+https://github.com/encode/starlette/blob/main/starlette/_utils.py
+"""
+
+from __future__ import annotations
+
+import functools
+import inspect
+from collections.abc import Awaitable, Callable
+from typing import Any, TypeGuard, TypeVar, overload
+
+T = TypeVar("T")
+
+AwaitableCallable = Callable[..., Awaitable[T]]
+
+
+@overload
+def is_async_callable(obj: AwaitableCallable[T]) -> TypeGuard[AwaitableCallable[T]]: ...
+
+
+@overload
+def is_async_callable(obj: Any) -> TypeGuard[AwaitableCallable[Any]]: ...
+
+
+def is_async_callable(obj: Any) -> Any:
+    while isinstance(obj, functools.partial):  # pragma: lax no cover
+        obj = obj.func
+
+    return inspect.iscoroutinefunction(obj) or (
+        callable(obj) and inspect.iscoroutinefunction(getattr(obj, "__call__", None))
+    )