Knowledge graphs and ontologies provide promising technical solutions for implementing the FAIR Principles for Findable, Accessible, Interoperable, and Reusable data and metadata. However, they also come with their own challenges. Nine such challenges are discussed and associated with the criterion of cognitive interoperability and specific FAIREr principles (FAIR + Explorability raised) that they fail to meet. We introduce an easy-to-use, open source knowledge graph framework that is based on knowledge graph building blocks (KGBBs). KGBBs are small information modules for knowledge-processing, each based on a specific type of semantic unit. By interrelating several KGBBs, one can specify a KGBB-driven FAIREr knowledge graph. Besides implementing semantic units, the KGBB Framework clearly distinguishes and decouples an internal in-memory data model from data storage, data display, and data access/export models. We argue that this decoupling is essential for solving many problems of knowledge management systems. We discuss the architecture of the KGBB Framework as we envision it, comprising (i) an openly accessible KGBB-Repository for different types of KGBBs, (ii) a KGBB-Engine for managing and operating FAIREr knowledge graphs (including automatic provenance tracking, editing changelog, and versioning of semantic units); (iii) a repository for KGBB-Functions; (iv) a low-code KGBB-Editor with which domain experts can create new KGBBs and specify their own FAIREr knowledge graph without having to think about semantic modelling. We conclude with discussing the nine challenges and how the KGBB Framework provides solutions for the issues they raise. While most of what we discuss here is entirely conceptual, we can point to two prototypes that demonstrate the principle feasibility of using semantic units and KGBBs to manage and structure knowledge graphs.