Previous research has used Large Language Models (LLMs) to develop personalized Conversational Recommender Systems (CRS) with text-based user interfaces (UIs). However, the potential of LLMs to generate interactive graphical elements that enhance user experience remains largely unexplored. To address this gap, we introduce "GenUI(ne) CRS," a novel framework designed to leverage LLMs for adaptive and interactive UIs. Our framework supports domain-specific graphical elements such as buttons and cards, in addition to text-based inputs. It also addresses the common LLM issue of outdated knowledge, known as the "knowledge cut-off," by implementing Retrieval-Augmented Generation (RAG). To illustrate its potential, we developed a prototype movie CRS. This work demonstrates the feasibility of LLM-powered interactive UIs and paves the way for future CRS research, including user experience validation, transparent explanations, and addressing LLM biases.