Extract the logic in a seperate class and call it directly. Use the cmdlet to be, well, just a shell around this new class.
This Seperation of Concerns (SoC) also enables easier unit tests and leads to an overall cleaner architecture.
  Extracted Class Greeter.cs
public class Greeter {
    public Greeter(string name) {
        _Name = name;
    }
    private string _Name;
    public string SayHello() {
        return $"Hello {_Name}";
    }
    public string SayGoodBye() {
        return $"So long {_Name}, and thanks for all the fish!";
    }
}
  CommandLet GetGreetingCommand.cs
[Cmdlet("Greeting", "Get")]
public class GetGreetingCommand : Cmdlet {
    [Parameter(Mandatory = true)]
    public string Name { get; set; }
    protected override void ProcessRecord() {
        var greeter = new Greeter(Name);
        var greeting = greeter.SayHello();
        WriteObject(greeting);
    }
}
  CommandLet GetGoodByeCommand .cs
[Cmdlet("GoodBye", "Get")]
public class GetGoodByeCommand : Cmdlet {
    [Parameter(Mandatory = true)]
    public string Name { get; set; }
    protected override void ProcessRecord() {
        var greeter = new Greeter(Name);
        var goodBye = greeter.SayGoodBye();
        WriteObject(goodBye);
    }
}
  Console Main.cs (or any other client-code of Greeter-class)
public static void main(string[] args) {
    var greeter = new Greeter(args.FirstOrDefault());
    Console.WriteLine(greeter.SayHello());
    Console.WriteLine(greeter.SayGoodBye());
}
  TestCase
public static void SayingHelloUsesName() {
    var sut = new Greeter("Arthur");
    var expected = "Hello Arthur";
    var actual = sut.SayHello();
    Assert.AreEqual(expected, actual);
}
The two concerns here are 
- the actual BusinessLogic (Greeter.cs)
- interoperability with PowerShell, providing mechanisms to parameterize the cmdlet, etc. (Get*Command.cs). As you see, the cmdlets really only pass through the calls, while enabling use via PowerShell.
@Mathias R. Jessen ´ answer could be usefull, if you need to call third party cmdlets, but in most cases, there should be an appropriate (non-powershell) API for what you are trying to do.