docs(claude_responses_api.md): update docs with mcp server tutorial

docs/my-website/docs/tutorials/claude_responses_api.md

@@ -209,4 +209,81 @@ claude --model claude-bedrock
 </TabItem>
 </Tabs>
 
-<Image img={require('../../img/release_notes/claude_code_demo.png')} style={{ width: '500px', height: 'auto' }} />
+<Image img={require('../../img/release_notes/claude_code_demo.png')} style={{ width: '500px', height: 'auto' }} />
+
+
+## Connecting MCP Servers
+
+You can also connect MCP servers to Claude Code via LiteLLM Proxy.
+
+:::note
+
+Limitations:
+
+- Currently, only HTTP MCP servers are supported
+- Does not work in Cursor IDE yet.
+
+:::
+
+1. Add the MCP server to your `config.yaml`
+
+In this example, we'll add the Github MCP server to our `config.yaml`
+
+```yaml title="config.yaml" showLineNumbers
+mcp_servers:
+  github_mcp:
+    url: "https://api.githubcopilot.com/mcp"
+    auth_type: oauth2
+    authorization_url: https://github.com/login/oauth/authorize
+    token_url: https://github.com/login/oauth/access_token
+    client_id: os.environ/GITHUB_OAUTH_CLIENT_ID
+    client_secret: os.environ/GITHUB_OAUTH_CLIENT_SECRET
+    scopes: ["public_repo", "user:email"]
+```
+
+2. Start LiteLLM Proxy
+
+```bash
+litellm --config /path/to/config.yaml
+
+# RUNNING on http://0.0.0.0:4000
+```
+
+3. Use the MCP server in Claude Code
+
+```bash
+claude mcp add --transport http litellm_proxy http://0.0.0.0:4000 --header "Authorization: Bearer sk-LITELLM_VIRTUAL_KEY"
+```
+
+4. Authenticate via Claude Code
+
+a. Start Claude Code
+
+```bash
+claude
+```
+
+b. Authenticate via Claude Code
+
+```bash
+/mcp
+```
+
+c. Select the MCP server
+
+```bash
+> litellm_proxy
+```
+
+d. Start Oauth flow via Claude Code
+
+```bash
+> 1. Authenticate
+ 2. Reconnect
+ 3. Disable             
+```
+
+e. Once completed, you should see this success message:
+
+<Image img={require('../../img/oauth_2_success.png')} style={{ width: '500px', height: 'auto' }} />
+

litellm/proxy/_experimental/mcp_server/mcp_server_manager.py

@@ -355,12 +355,12 @@ class MCPServerManager:
                     )
 
                     # Update tool name to server name mapping (for both prefixed and base names)
-                    self.tool_name_to_mcp_server_name_mapping[
-                        base_tool_name
-                    ] = server_prefix
-                    self.tool_name_to_mcp_server_name_mapping[
-                        prefixed_tool_name
-                    ] = server_prefix
+                    self.tool_name_to_mcp_server_name_mapping[base_tool_name] = (
+                        server_prefix
+                    )
+                    self.tool_name_to_mcp_server_name_mapping[prefixed_tool_name] = (
+                        server_prefix
+                    )
 
                     registered_count += 1
                     verbose_logger.debug(
@@ -1276,7 +1276,7 @@ class MCPServerManager:
             get_prisma_client_or_throw,
         )
 
-        verbose_logger.info("Loading MCP servers from database into registry...")
+        verbose_logger.debug("Loading MCP servers from database into registry...")
 
         # perform authz check to filter the mcp servers user has access to
         prisma_client = get_prisma_client_or_throw(
@@ -1292,7 +1292,9 @@ class MCPServerManager:
             )
             self.add_update_server(server)
 
-        verbose_logger.info(f"Registry now contains {len(self.get_registry())} servers")
+        verbose_logger.debug(
+            f"Registry now contains {len(self.get_registry())} servers"
+        )
 
     def get_mcp_server_by_id(self, server_id: str) -> Optional[MCPServer]:
         """

litellm/proxy/_new_secret_config.yaml

@@ -1,4 +1,14 @@
 model_list:
-  - model_name: gpt-5-codex
+  - model_name: gpt-5-mini
     litellm_params:
-      model: gpt-5-codex
+      model: gpt-5-mini
+
+mcp_servers:
+  github_mcp:
+    url: "https://api.githubcopilot.com/mcp"
+    auth_type: oauth2
+    authorization_url: https://github.com/login/oauth/authorize
+    token_url: https://github.com/login/oauth/access_token
+    client_id: os.environ/GITHUB_OAUTH_CLIENT_ID
+    client_secret: os.environ/GITHUB_OAUTH_CLIENT_SECRET
+    scopes: ["public_repo", "user:email"]