To help users do complex work, researchers have developed techniques to integrate AI and human intelligence into user interfaces (UIs). With the recent introduction of large language models (LLMs), which can generate text in response to a natural language prompt, there are new opportunities to consider how to integrate LLMs into UIs. We present Prompt Middleware, a framework for generating prompts for LLMs based on UI affordances. These include prompts that are predefined by experts (static prompts), generated from templates with fill-in options in the UI (template-based prompts), or created from scratch (free-form prompts). We demonstrate this framework with FeedbackBuffet, a writing assistant that automatically generates feedback based on a user's text input. Inspired by prior research showing how templates can help non-experts perform more like experts, FeedbackBuffet leverages template-based prompt middleware to enable feedback seekers to specify the types of feedback they want to receive as options in a UI. These options are composed using a template to form a feedback request prompt to GPT-3. We conclude with a discussion about how Prompt Middleware can help developers integrate LLMs into UIs.