You have linked the processes together with start_link. If the game crashes, it will also call whatever process is calling Game.Monitor.run. You should start the game under a supervisor.

Thanks for answer. I created a GenServer responsible of the Game monitoring and it works !

Here is the code if anyone reading this later needs it

defmodule Game.Monitor do
  @moduledoc """
  A simple GenServer used to monitor a game process
  """

  use GenServer
  require Logger

  def init(channel_id) do
    # On init try to attach to an existing game process
    # In case of Monitor failure, this will reattach the monitor to the game process
    if Process.whereis(Game) do
      Process.monitor(Game)
    end

    {:ok, channel_id}
  end

  def start_link(channel_id) do
    GenServer.start_link(__MODULE__, channel_id, name: __MODULE__)
  end

  def monitor() do
    GenServer.cast(__MODULE__, :monitor)
  end

  def handle_cast(:monitor, channel_id) do
    Process.monitor(Game)
    {:noreply, channel_id}
  end

  def handle_info({:DOWN, _ref, :process, object, reason}, channel_id) when reason != :killed do
    Logger.info("Game process monitor detected a game crash", reason: reason, data: object)

    Nostrum.Api.create_message!(
      channel_id,
      embed: %Nostrum.Struct.Embed{
        :title => "KABOOM ALERT, I just explode, sorry.",
        :description => "💥",
        :color => Colors.get_color(:danger)
      }
    )

    {:noreply, channel_id}
  end
end

I now need to make it less static but I get the idea. Thaks a lot.