Studying the gene regulatory networks (GRNs) that govern how cells change into specific cell types with unique roles throughout development is an active area of experimental research. The fate specification process can be viewed as a program prescribing the system dynamics, governed by a network of genetic interactions. To investigate the possibility that GRNs are not fixed but rather change their topology, for example as cells progress through commitment, we introduce the concept of Switching Gene Regulatory Networks (SGRNs) to enable the modelling and analysis of network reconfiguration. We define the synthesis problem of constructing SGRNs that are guaranteed to satisfy a set of constraints representing experimental observations of cell behaviour. We propose a solution to this problem that employs methods based upon Satisfiability Modulo Theories (SMT) solvers, and evaluate the feasibility and scalability of our approach by considering a set of synthetic benchmarks exhibiting possible biological behaviour of cell development. We outline how our approach is applied to a more realistic biological system, by considering a simplified network involved in the processes of neuron maturation and fate specification in the mammalian cortex.