refactor(event-handler): move _registered_api_adapter_async to async_utils.py

Per maintainer feedback on #8157 - function belongs in async_utils.py alongside other async event handler internals (wrap_middleware_async, _run_sync_middleware_in_thread, etc.)

Signed-off-by: hirenkumar-n-dholariya <hirenkumarnd@gmail.com>

Author: hirenkumar-n-dholariya

aws_lambda_powertools/event_handler/api_gateway.py

@@ -3,7 +3,6 @@
 import base64
 import json
 import logging
-import inspect
 import re
 import traceback
 import warnings
@@ -1475,64 +1474,7 @@ def _registered_api_adapter(
 
     return app._to_response(next_middleware(**route_args))
 
-async def _registered_api_adapter_async(
-    app: ApiGatewayResolver,
-    next_middleware: Callable[..., Any],
-) -> dict | tuple | Response | BedrockResponse:
-    """
-    Async version of _registered_api_adapter.
-
-    Detects if the route handler is a coroutine and awaits it.
-    _to_response() stays sync (CPU-bound — no async benefit).
-
-    **IMPORTANT: This is an internal building block only.
-    Nothing calls it in the resolve chain yet. It will be used
-    by resolve_async() (see issue #8137).
-
-    Parameters
-    ----------
-    app: ApiGatewayResolver
-        The API Gateway resolver
-    next_middleware: Callable[..., Any]
-        The function to handle the API
-
-    Returns
-    -------
-    Response
-        The API Response Object
-    """
-    route_args: dict = app.context.get("_route_args", {})
-    logger.debug(f"Calling async API Route Handler: {route_args}")
-
-    # Inject a Request object when the handler declares a parameter typed as Request.
-    # Lookup is cached on the Route object to avoid repeated signature inspection.
-    route: Route | None = app.context.get("_route")
-    if route is not None:
-        if not route.request_param_name_checked:
-            route.request_param_name = _find_request_param_name(next_middleware)
-            route.request_param_name_checked = True
-        if route.request_param_name:
-            route_args = {**route_args, route.request_param_name: app.request}
-
-        # Resolve Depends() parameters (same as sync version)
-        if route.has_dependencies:
-            from aws_lambda_powertools.event_handler.depends import build_dependency_tree, solve_dependencies
-
-            dep_values = solve_dependencies(
-                dependant=build_dependency_tree(route.func),
-                request=app.request,
-                dependency_overrides=app.dependency_overrides or None,
-            )
-            route_args.update(dep_values)
-
-    # Call handler — detect if result is a coroutine and await it
-    result = next_middleware(**route_args)
-    if inspect.iscoroutine(result):
-        result = await result
 
-    # _to_response is CPU-bound, stays sync
-    return app._to_response(result)
-    
 class ApiGatewayResolver(BaseRouter):
     """API Gateway, VPC Lattice, Bedrock and ALB proxy resolver