Laser wakefield accelerators produce accelerating gradients up to hundreds of GeV/m, and recently demonstrated 1-10 MeV energy spread at energies up to 1 GeV using electrons self-trapped from the plasma. Controlled injection and staging may further improve beam quality by circumventing tradeoffs between energy, stability, and energy spread/emittance. We present experiments demonstrating production of a stable electron beam near 1 MeV with hundred-keV level energy spread and central energy stability by using the plasma density profile to control selfinjection, and supporting simulations. Simulations indicate that such beams can be post accelerated to high energies,potentially reducing momentum spread in laser acceleratorsby 100-fold or more.