With the aim to provide teachers with more specific, frequent, and actionable feedback about their teaching, we explore how Large Language Models (LLMs) can be used to estimate "Instructional Support" domain scores of the CLassroom Assessment Scoring System (CLASS), a widely used observation protocol. We design a machine learning architecture that uses either zero-shot prompting of Meta's Llama2, and/or a classic Bag of Words (BoW) model, to classify individual utterances of teachers' speech (transcribed automatically using OpenAI's Whisper) for the presence of Instructional Support. Then, these utterance-level judgments are aggregated over a 15-min observation session to estimate a global CLASS score. Experiments on two CLASS-coded datasets of toddler and pre-kindergarten classrooms indicate that (1) automatic CLASS Instructional Support estimation accuracy using the proposed method (Pearson R up to 0.48) approaches human inter-rater reliability (up to R = 0.55); (2) LLMs generally yield slightly greater accuracy than BoW for this task, though the best models often combined features extracted from both LLM and BoW; and (3) for classifying individual utterances, there is still room for improvement of automated methods compared to human-level judgments. Finally, (4) we illustrate how the model's outputs can be visualized at the utterance level to provide teachers with explainable feedback on which utterances were most positively or negatively correlated with specific CLASS dimensions.