Picture descriptions provide valuable insights into several clinical constructs related to cognitive-linguistic abilities. However, operationalizing these constructs into quantitative measures remains challenging, limiting interpretability and clinical utility. We introduced seven constructs tailored to the Cookie Theft picture description task and prompted large language models (LLMs) to evaluate them, generating severity scores and example-based explanations. Among the examined LLMs, Claude 3.5 Sonnet performed the best, producing severity scores that significantly distinguish cognitively impaired individuals from healthy controls. The model achieves a high accuracy of 85% on the ADReSS dataset. Expert evaluation of Claude's scores and explanations yields a 3.99/5 average agreement. The findings demonstrate the potential of LLMs to operationalize clinical constructs and generate interpretable evaluations, offering a promising approach for accessible cognitive screening tools.