As alternatives to the used bike/vintage resto-mod route, which is a great way to go if you know what you're doing and have the patience, Diamondback has some good budget options; Century 2 ($850) or Hanjo 3 ($1000) would make fine brevet bikes. For something a little more traditional, the
DIY Velo Routier ($1400) is a decent option. In that same price range, there are also a few of off-the-peg touring bikes (Jamis, Surley, Kona) that plenty of people including myself have used to get started in randonneuring.