Large language models (LLMs) promise to accelerate UI design, yet current tools struggle with two fundamentals: externalizing designers' intent and controlling iterative change. We introduce SPEC, a structured, parameterized, hierarchical intermediate representation that exposes UI elements as controllable parameters. Building on SPEC, we present SpecifyUI, an interactive system that extracts SPEC from UI references via region segmentation and vision-language models, composes UIs across multiple sources, and supports targeted edits at global, regional, and component levels. A multi-agent generator renders SPEC into high-fidelity designs, closing the loop between intent expression and controllable generation. Quantitative experiments show SPEC-based generation more faithfully captures reference intent than prompt-based baselines. In a user study with 16 professional designers, SpecifyUI significantly outperformed Stitch on intent alignment, design quality, controllability, and overall experience in human-AI co-creation. Our results position SPEC as a specification-driven paradigm that shifts LLM-assisted design from one-shot prompting to iterative, collaborative workflows.