Trying to automatically rebuild and restart .NET console application on linux

Question

I'm making a small webserver .NET console application on my windows machine that will run on a linux server (amazon os), and I'd like to make it automatically update itself when I push new code to github.

Automatically pulling from github already works by waiting for a webhook and then creating a Process which, as I understand it, practically lets me run a command line from the code.

using System.Diagnostics;

public static class GitApi
{
    public static void Pull(string path)
    {
        Process process = new Process()
        {
            StartInfo = new ProcessStartInfo()
                {
                FileName = "git",
                Arguments = "pull",
                UseShellExecute = false,
                RedirectStandardOutput = true,
                WorkingDirectory = path
            }
        };
        process.Start();
    }
}

What I'd then like to do is to automatically rebuild and restart after pulling from git, by using the dotnet run command in a similar way.

using System.Diagnostics;

public static class Restarter
{
    public static void Restart(string path)
    {
        Process process = new Process()
        {
            StartInfo = new ProcessStartInfo()
            {
                FileName = "dotnet",
                Arguments = "run",
                UseShellExecute = false,
                WorkingDirectory = path
            }
        };
        process.Start();
        Environment.Exit(0);
    }
}

While this seems to work fine on windows, when I try on linux it seems to exit the main process and then start the new one in the background somehow. It still writes in the same screen, but I can't kill it with Ctrl + C. Instead I have to find the process by checking the port that it's using and then use the kill command.

Could I fix this somehow or is there another simple way to rebuild and restart a .NET console application?