Fuzzing is a well-established technique in the software domain to uncover bugs and vulnerabilities. Yet, applications of fuzzing for security vulnerabilities in hardware systems are scarce, as principal reasons are requirements for design information access (HDL source code). Moreover, observation of internal hardware state during runtime is typically an ineffective information source, as its documentation is often not publicly available. In addition, such observation during runtime is also inefficient due to bandwidth-limited analysis interfaces (JTAG, and minimal introspection of internal modules). In this work, we investigate fuzzing for 7-Series and UltraScale(+) FPGA configuration engines, the control plane governing the (secure) bitstream configuration within the FPGA. Our goal is to examine the effectiveness of fuzzing to analyze and document the opaque inner workings of FPGA configuration engines, with a primary emphasis on identifying security vulnerabilities. Using only the publicly available chip and dispersed documentation, we first design and implement ConFuzz, an advanced FPGA configuration engine fuzzing and rapid prototyping framework. Based on our detailed understanding of the bitstream file format, we then systematically define 3 novel key fuzzing strategies for Xilinx configuration engines. Moreover, our strategies are executed through mutational structure-aware fuzzers and incorporate various novel custom-tailored, FPGA-specific optimizations. Our evaluation reveals previously undocumented behavior within the configuration engine, including critical findings such as system crashes leading to unresponsive states of the FPGA. In addition, our investigations not only lead to the rediscovery of the starbleed attack but also uncover JustSTART (CVE-2023-20570), capable of circumventing RSA authentication for Xilinx UltraScale(+). Note that we also discuss countermeasures.