Following an API-led, design-first approach, the first step is to update the API's contract. The RAML specification must be modified to include the new resource method (e.g., PUT or PATCH) for updating quotes. This defines the new functionality for consumers.

Subsequently, the API implementation, which contains the business logic, must be updated. The Mule application needs to be modified to handle requests to this new endpoint, process the update logic against the quoting engine, and return the appropriate response as defined in the updated RAML. These two steps are fundamental to introducing new functionality to an existing API.

B. Remove the old client applications and create new client applications to account for the changes: This is incorrect. Adding a new method is a non-breaking change; existing clients that only create quotes will continue to function without modification.

D. Deprecate existing versions of the API in Exchange: Deprecation is used to retire an API version, not to add functionality. This change would likely result in a minor version update, not the deprecation of the current one.

E. Add a new API policy to API Manager to allow access to the updated endpoint: Applying policies is a governance step performed after the API is implemented and deployed. It is not a core requirement to enable the processing of the new functionality itself.

1. MuleSoft Documentation, API-Led Connectivity: This foundational concept outlines the process of designing APIs first (the contract) and then building the implementation. The scenario requires updating both the design (RAML) and the implementation. (Source: MuleSoft Official Documentation, "API-Led Connectivity" whitepaper).

2. MuleSoft Documentation, API Development Lifecycle -> Design an API: "The first step in the API lifecycle is to design the API. In this phase, you use a specification language such as RAML or OAS to define the API’s resources, methods, parameters, and responses." This directly supports updating the RAML (Option C) as the initial step.

3. MuleSoft Documentation, API Development Lifecycle -> Develop an API implementation: "After you design an API specification, you can create an application that implements the specification. You can scaffold an API implementation from an API specification in Anypoint Studio." This supports the need to update the implementation (Option A) to match the new design.