How do I set a readonly field in an initialize method that gets called from the constructor?

Rob’s answer is the way to do it, in my book. If you need to initialize multiple fields you can do it using out parameters:

public class Class
{
    private readonly int readonlyField1;
    private readonly int readonlyField2;

    public Class()
    {
        Init(out readonlyField1, out readonlyField2);
    }

    protected virtual void Init(out int field1, out int field2)
    {
        field1 = 1;
        field2 = 2;
    }
}

Personally I find this makes sense in certain scenarios, such as when you want your fields to be readonly but you also want to be able to set them differently in a derived class (without having to chain a ton of parameters through some protected constructor). But maybe that’s just me.