Is your feature request related to a problem? Please describe.
With the release of v1.9.0, the fast_api.py module was refactored to allow simpler construction of custom API servers.
Currently, the documentation does not provide a reference implementation or example usage showing how developers can extend these new modular capabilities in practice.
This makes it harder for teams to quickly adopt the new design, especially when:

• Adding custom routers
• Overriding ADK endpoints
• Optimizing SSE streaming
• Enabling hot-reload during development

Describe the solution you'd like
Add a contributed example project that demonstrates:

• A modular FastAPI server implementation built on top of the v1.9.0 changes
• Example project structure (fastapi_modular_server/) with routers, agents, config, and utilities
• Optimized SSE event mapper with multiple streaming modes
• Agent hot-reload example
• A README.md that walks through extending ADK servers

This example would serve as a starting point/template for developers building production-ready servers without modifying ADK core.

Describe alternatives you've considered

• Inline code snippets in the docs
• Relying on release notes alone

Additional context
This request is based on the following release item:

[CLI] Modularize fast_api.py to allow simpler construction of API Server ([bfc203a](bfc203a), [dfc25c1](dfc25c1), [e176f03](e176f03))