The short answer is that AI image generation just isn't quite there yet. (This probably isn't true in the near future.)
Anyone who is dabbling with image generation has this problem of how to gain control. When people post a nice and seemingly perfect generated image, we can never know if the image really is exactly what the creator had in mind or if they just adjusted their expectations after seeing the result.
Midjourney makes nice images out of the box, but perhaps lacks fine grained control. I'm not sure. I haven't used it much to be honest. I've mostly played around with Stable Diffusion. It evolves chaotically and fast. So it's less accessible, but it has more possibilities to customize.
Here's a list of things you could consider to refine your images. Some of them are only available in Stable Diffusion but some of them are general advice:
Keep refining your prompt (and negative prompt) to get closer to the result you want. Be creative. Sometimes you need to find a balance between positive and negative prompt words and sometimes it's not exactly logical which words to use. You have to experiment. (You might get an image of a person looking too young. Adding old to the prompt might change the whole mood of the image whereas adding zombie with a low weight might just make the person a little darker around the eyes, achieving your goal. I have no idea if it would work, but perhaps in your case trying words like wizard or hermit might provoke a long beard?)
Make larger quantities. You are talking about generating four images and then four more. Perhaps you need to generate hundreds of images and then search for the needle in the haystack.
Use inpaint to change parts of your image. This is an iterative process which might require many generations before you are satisfied with the result.
Use Photoshop or similar to manually edit your image. (This is probably the real trick to many of the "perfect" generations we see online.)
Use something like InstructPix2Pix (https://github.com/timothybrooks/instruct-pix2pix) or similar. It's months old so something better has probably emerged. It aims to make it possible to adjust an image using a prompt. So in theory you should be able to prompt make the beard longer. In my experience it doesn't work so well, but it's probably only a matter of time.
Use ControlNet (https://github.com/lllyasviel/ControlNet). It's not exactly what you are asking for, but it makes it possible to use different aspects of the composition of an input image to generate a new image. Lots of possibilities for creative iterative use.
(The Stable Diffusion subreddit is a good place to get updated on the latest development and get tips and tricks: https://www.reddit.com/r/StableDiffusion/)
(Most people using Stable Diffusion use Automatic1111's Stable Diffusion webui: https://github.com/AUTOMATIC1111/stable-diffusion-webui)
