We present novel extensions for combining frameworks that quantify both aleatoric (i.e., irreducible) and epistemic (i.e., reducible) sources of uncertainties in the modelling and design of engineered systems. The data-consistent (DC) framework [1,2,3] poses an inverse problem and solution for quantifying aleatoric uncertainties in terms of pullback and push-forward measures for a given Quantity of Interest (QoI) map. Unfortunately, a pre-specified QoI map is not always available a priori to the collection of data associated with system outputs. The data themselves are often polluted with measurement errors (i.e., epistemic uncertainties), which complicates the process of specifying a useful QoI. The Learning Uncertain Quantities (LUQ) framework [4,5] defines a formal three-step machine-learning enabled process for transforming noisy datasets into samples of a learned QoI map to enable DC-based inversion. We summarize the combination of these frameworks to quantify uncertainties in problems relevant to engineering design and manufacturing. We then discuss how the underlying mathematical theory can help address fundamental issues around optimal data acquisition efforts when a digital twin is incorporated. Such issues involve identifying where and when to collect data as well as how much data is numerically sufficient for quantifying uncertainties.