Fast machine code generation is especially important for fast start-up just-in-time compilation, where the compilation time is part of the end-to-end latency. However, widely used compiler frameworks like LLVM do not prioritize fast compilation and require an extra IR translation step increasing latency even further; and rolling a custom code generator is a substantial engineering effort, especially when targeting multiple architectures. Therefore, in this paper, we present TPDE, a compiler back-end framework that adapts to existing code representations in SSA form. Using an IR-specific adapter providing canonical access to IR data structures and a specification of the IR semantics, the framework performs one analysis pass and then performs the compilation in just a single pass, combining instruction selection, register allocation, and instruction encoding. The generated target instructions are primarily derived code written in high-level language through LLVM's Machine IR, easing portability to different architectures while enabling optimizations during code generation. To show the generality of our framework, we build a new back-end for LLVM from scratch targeting x86-64 and AArch64. Performance results on SPECint 2017 show that we can compile LLVM-IR 8--24x faster than LLVM -O0 while being on-par in terms of run-time performance. We also demonstrate the benefits of adapting to domain-specific IRs in JIT contexts, particularly WebAssembly and database query compilation, where avoiding the extra IR translation further reduces compilation latency.