I don't know of one that exists, but there's no reason one couldn't, or shouldn't.
Writing a compiler for a C-like language is a classic project for one-semester college compiler courses. If you know C# already, it provides a lot of features which will make your job easier than when I was in college! There are plenty of libraries sitting around which will make the job easier, without taking away the challenge, and you can always replace them with your own ad-hoc code if you need flexibility they don't provide.
The first C compiler was written in BCPL because it's what they had. Current C compilers are usually written in C because they aim to be portable. I don't think anyone would argue that C is a good language for writing compilers in. (C# isn't perfect but it's a lot better!) In a statically-compiled language like C, I don't think you get much benefit, if any, from using the target language to write the compiler.
A compiler in an HLL potentially has many advantages. It'd be shorter and simpler than one written in C. That alone could make a lot of things sufficiently easier that they could be pulled below the threshold of "too hard that nobody's ever going to do them". (GCC is kind of the poster-child for how a compiler written in a LLL can be so complex that it moves at a glacial rate.) Optimizations are basically graph transformations, which aren't exactly C's forte.
I don't consider it "backwards" at all to use C# to compile C. Unless somebody's proposing to rewrite all their C code in a higher-level language, it still needs to be compiled somehow, and that means you need a compiler. Shouldn't that compiler be written with tools that enable it to offer the best reliability and performance?
Good luck! I look forward to seeing what you write!