Optical coherence tomography (OCT) is a prevalent, interferometric, high-resolution imaging method with broad biomedical applications. Nonetheless, OCT images suffer from an artifact called speckle, which degrades the image quality. Digital filters offer an opportunity for image improvement in clinical OCT devices, where hardware modification to enhance images is expensive. To reduce speckle, a wide variety of digital filters have been proposed; selecting the most appropriate filter for an OCT image/image set is a challenging decision, especially in dermatology applications of OCT where a different variety of tissues are imaged. To tackle this challenge, we propose an expandable learnable despeckling framework, we call LDF. LDF decides which speckle reduction algorithm is most effective on a given image by learning a figure of merit (FOM) as a single quantitative image assessment measure. LDF is learnable, which means when implemented on an OCT machine, each given image/image set is retrained and its performance is improved. Also, LDF is expandable, meaning that any despeckling algorithm can easily be added to it. The architecture of LDF includes two main parts: (i) an autoencoder neural network and (ii) filter classifier. The autoencoder learns the FOM based on several quality assessment measures obtained from the OCT image including signal-to-noise ratio, contrast-to-noise ratio, equivalent number of looks, edge preservation index, and mean structural similarity index. Subsequently, the filter classifier identifies the most efficient filter from the following categories: (a) sliding window filters including median, mean, and symmetric nearest neighborhood, (b) adaptive statistical-based filters including Wiener, homomorphic Lee, and Kuwahara, and (c) edge preserved patch or pixel correlation-based filters including nonlocal mean, total variation, and block matching three-dimensional filtering.