This work studies the applicability of expensive external oracles such as large language models in answering top-k queries over predicted scores. Such scores are incurred by user-defined functions to answer personalized queries over multi-modal data. We propose a generic computational framework that handles arbitrary set-based scoring functions, as long as the functions could be decomposed into constructs, each of which sent to an oracle (in our case an LLM) to predict partial scores. At a given point in time, the framework assumes a set of responses and their partial predicted scores, and it maintains a collection of possible sets that are likely to be the true top-k. Since calling oracles is costly, our framework judiciously identifies the next construct, i.e., the next best question to ask the oracle so as to maximize the likelihood of identifying the true top-k. We present a principled probabilistic model that quantifies that likelihood. We study efficiency opportunities in designing algorithms. We run an evaluation with three large scale datasets, scoring functions, and baselines. Experiments indicate the efficacy of our framework, as it achieves an order of magnitude improvement over baselines in requiring LLM calls while ensuring result accuracy. Scalability experiments further indicate that our framework could be used in large-scale applications.