Advancements in design automation technologies, such as high-level synthesis (HLS), have raised the input abstraction level and made the design entry process for FPGAs more friendly to software programmers. In contrast, the backend compilation process for implementing designs on FPGAs is considerably more lengthy compared to software compilation: while software code compilation may take just a few seconds, FPGA compilation times can often span from several minutes to hours due to the complexity of the underlying toolchain and ever-growing device capacities. In this paper, we present DynaRapid, a fast compilation tool that generates - in a matter of seconds - fully legal placed-and-routed designs for commercial FPGAs. Elastic circuits created by the HLS tool Dynamatic are made exclusively of a limited number of reusable components; we exploit this fact to create a library of placed and routed building blocks, and then stitch together instances of them as needed through RapidWright. Our approach accelerates the C-to-FPGA implementation process by a geomean 20 × with only 10% of degradation in operating frequency compared to a conventional commercial off-the-shelf implementation flow.