# [Charcoal], <s>30</s> 29 bytes

    ＮθＮηＩＥＸθηＥ﹪÷ιＸθ⮌…⁰η⊗θ⌊⟦⁻⊖⊗θλλ

[Try it online!][TIO-l0jf17wv] Link is to verbose version of code. Outputs each digit of the gray code as a base 10 number on its own line, with each gray code double-spaced from the next. Explanation:

    Ｎθ                              Input `N` as a number
      Ｎη                            Input `k` as a number
           θ                        Input `N`
          Ｘ                         To the power of
            η                       Input `k`
         Ｅ                          Map over implicit range
                ι                   Current value
               ÷                    Vectorised integer divide
                  θ                 Input `N`
                 Ｘ                  Vectorised to power
                    …               Range from
                     ⁰              Literal integer `0`
                      η             To input `k`
                   ⮌                Reversed
              ﹪                     Vectorised modulo
                        θ           Input `N`
                       ⊗            Doubled
             Ｅ                      Map over list
                              θ     Input `N`
                             ⊗      Doubled
                            ⊖       Decremented
                           ⁻        Subtract
                               λ    Current value
                         ⌊⟦         Minimum of that and
                                λ   Current value
              ⮌                     Reversed
        Ｉ                           Cast to string
                                    Implicitly print

If omission of leading zeros is allowed then for 28 bytes:

    ＮθＮηＩＥＥＸθη↨ιθＥι⎇﹪↨…ιμθ²⁻⊖θλλ

[Try it online!][TIO-l0jeexrb] Link is to verbose version of code. Explanation:

    Ｎθ                              Input `N` as a number
      Ｎη                            Input `k` as a number
            θ                       Input `N`
           Ｘ                        To the power of
             η                      Input `k`
          Ｅ                         Map over implicit range
               ι                    Current value
              ↨                     Converted to base
                θ                   Input `N`
         Ｅ                          Map over base `N` values
                  ι                 Current base `N` value
                 Ｅ                  Map over base `N` digits
                      …ιμ           Current prefix
                     ↨              Convert from base
                         θ          Input `N`
                    ﹪               Modulo
                          ²         Literal integer `2`
                   ⎇                If odd then
                             θ      Input `N`
                            ⊖       Decremented
                           ⁻        Subtract
                              λ     Current digit
                               λ    Else current digit
        Ｉ                           Cast to string
                                    Implicitly print

(Note that *all* leading zeros are omitted, so the first gray code is empty.)

[Charcoal]: https://github.com/somebody1234/Charcoal
[TIO-l0jf17wv]: https://tio.run/##TYzNCoMwEITvfYocV0ihiDeP9eJBEekLbHXRQEzMGi19@jQp9GdhF2a/mRlm5MGiDqE26@7bfbkTg8vK07@eo@5YGQ9X3Dw0dty1hQZX6Owj@aWYMynSoza@UocaCZQUX9rTQbwR9Ggmgkuyx5HiRmyQn5/GX1hL4SLP41Y0MC1kPI3Q0oQ@0pTV7wYXbxlCIYpwPvQL "Charcoal – Try It Online"
[TIO-l0jeexrb]: https://tio.run/##TYu9DsMgEIP3PgXjIdGlypatdOmQKkNf4EpOAolAwk@rPD0FutSSJVv@rDQG5dGWcndbTo@8vijAzsfTf9e1z8G4BBJjggm37tl/GiyY5oJdMRIYwfaa21jjk4LDcMDkl2w9dEIeypLUvgMr//GX9jEuR7iRCrSSS7RAW2x301jKwIZyftsv "Charcoal – Try It Online"