Emacs

Check the documentation of async-shell-command (M-x describe-function async-shell-command RET), and you'll see that it takes one necessary argument, the command to run.

If you run it interactively (through its keybinding or M-x), it prompts you for that argument. Type in the command for Emacs, and Emacs will pass that as the argument to async-shell-command.

But if you want to, you can call it from Lisp and pass in the argument yourself! Try executing it in the scratch buffer or by running eval-expression (I believe M-: by default).

(async-shell-command "git status")

That will run git status and give you the result

Now if you want to run that bit of code more often, wrap it in a function and assign it a keybinding!

(defun crj-check-git-status ()
  (interactive)
  (async-shell-command "git status"))

(general-define-key (kbd "C-M-&") #'crj-check-git-status)

Would anyone else handle this differently? I'm still learning myself!

You could make your own wrappers around async-shell-command and shell-command to autofill with your last command + have your whole history. Try the code below to see if it does the job like you'd like.

(defvar async-shell-command-history nil
  "List of commands executed with `async-shell-command`.")

(defvar shell-command-history nil
  "List of commands executed with `shell-command`.")

(defadvice async-shell-command (before save-command-in-history activate)
  "Save the command executed with `async-shell-command` in its history."
  (let ((command (ad-get-arg 0)))
    (unless (string-blank-p command)
      (setq async-shell-command-history (cons command async-shell-command-history)))))

(defadvice shell-command (before save-command-in-history activate)
  "Save the command executed with `shell-command` in its history."
  (let ((command (ad-get-arg 0)))
    (unless (string-blank-p command)
      (setq shell-command-history (cons command shell-command-history)))))

(defun my-async-shell-command ()
  "Run `async-shell-command` with a choice from its command history."
  (interactive)
  (let* ((command (completing-read "Async shell command: "
                                   async-shell-command-history
                                   nil nil
                                   (car async-shell-command-history)))
         (final-command (if (string-blank-p command) 
                            (or (car async-shell-command-history) "")
                          command)))
    (async-shell-command final-command)))

(defun my-shell-command ()
  "Run `shell-command` with a choice from its command history."
  (interactive)
  (let* ((command (completing-read "Shell command: "
                                   shell-command-history
                                   nil nil
                                   (car shell-command-history)))
         (final-command (if (string-blank-p command) 
                            (or (car shell-command-history) "")
                          command)))
    (shell-command final-command)))

Well done finding an answer.

I tweaked the answer somewhat, as it bothered me that we were getting the most recent command inserted and having it show up again when you decide to go back to an earlier command. In other words, with the OP's advice, we get the most recent command in the minibuffer history twice, once auto-filled and once in the regular history.

I'm sure there are cleaner ways to do it than using a custom global variable no other functions know about, which could I think lead to some weird side effects at some point... but for now, I like the behavior:

(defvar crj-most-recent-shell-command (car shell-command-history))

(defun crj--pop-shell-command-history (fn &rest _args)
  (setq crj-most-recent-shell-command (car shell-command-history))
  (let ((shell-command-history (cdr shell-command-history)))
    (apply fn _args)))

(defun crj--auto-fill-shell-commands (args)
  (list (car args) crj-most-recent-shell-command))

(advice-add 'read-shell-command :around #'crj--pop-shell-command-history)
(advice-add 'read-shell-command :filter-args #'crj--auto-fill-shell-commands)

Let me know if anyone has improvements/objections!

Slight improvement on your advising function, OP: this version starts the shell history you scroll through at index 1, so that after you go up one from your most recent-command, you don't see the most recent command again.

(defun crj--auto-fill-shell-commands (args)
  (list (car args) (car shell-command-history) '(shell-command-history . 1) (cdddr args)))

(advice-add 'read-shell-command :filter-args #'crj--auto-fill-shell-commands)

Is there a cleaner way to handle the args than this unpleasing cdddr? Something I can do with &rest?