Confusion with `WWW-Authenticate` and EXTERNAL server mode

Describe the bug
It seems there is some confusion in this example project regarding the EXTERNAL mode.

In this mode, there should be two distinct servers:

• one for authentication (OAuth2),
• one for the MCP server.

However, even in EXTERNAL mode, the example project still exposes the resource /.well-known/oauth-authorization-server on the MCP server.
This should not be the case. This resource is exposed by the OAuth2 authorization server, not by the MCP server, no ?

According to the MCP specification, the MCP server should instead expose: /.well-known/oauth-protected-resource
(see: https://modelcontextprotocol.io/specification/2025-11-25/basic/authorization#protected-resource-metadata-discovery-requirements)

This resource is discovered by the client via the WWW-Authenticate header, which contains the URL of the MCP protected-resource metadata.

The JSON returned by this /.well-known/oauth-protected-resource endpoint must provide the base URL of the external authorization server.
Example response:

{
  "resource": "http://localhost:3232/mcp",
  "resource_name": "MCP Server",
  "authorization_servers": [
    "http://localhost:3001"
  ]
}

The client then uses this information to discover the authorization server metadata, as described in the spec:
https://modelcontextprotocol.io/specification/2025-11-25/basic/authorization#authorization-server-metadata-discovery

The spec allows several discovery patterns, such as:

• OAuth 2.0 Authorization Server Metadata with path insertion:
  https://auth.example.com/.well-known/oauth-authorization-server/tenant1

• OpenID Connect Discovery 1.0 with path insertion:
  https://auth.example.com/.well-known/openid-configuration/tenant1

• OpenID Connect Discovery 1.0 with path appending:
  https://auth.example.com/tenant1/.well-known/openid-configuration

This seems to be related to this topic on the inspector. Which seems to ignore the WWW-Authenticate header and make calls directly to the MCP server, expecting to deal with the authentication server.

To Reproduce

This issue can be observed by testing MCP clients + server that rely on this discovery mechanism.
For these clients, the current implementation does not work as expected.

Expected behavior

Add the /.well-known/oauth-protected-resource endpoint to the MCP server.
Remove the .well-known/oauth-authorization-server on the Mcp Server.Because it already displays this one by default.

Logs
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Additional context
Add any other context about the problem here.

Bắt đầu từ cấu hình route well-known của MCP server và kiểm tra cách chế độ EXTERNAL expose /.well-known/oauth-authorization-server. So sánh hành vi với MCP authorization specification được liên kết và xác minh flow WWW-Authenticate. Được xem là hoàn tất khi MCP server expose /.well-known/oauth-protected-resource cùng URL của authorization server bên ngoài và không còn expose endpoint metadata của authorization server.