Add icon to module's menu item in Administration menu
I figured it out. Now, this might not be the very best way to solve it, and some things should probably be changed, but this will be fine for now. Let's assume the module name is pizza.

First, create a directory called css. Inside this folder, create a css file with the name of the module. In this case it would be pizza.css.
Since there needs to be an icon next to the module's menu item title on the administration menu, this is the CSS we'll need (or similar):

.toolbar-icon-pizza-main::before{
    background-image: url('../../../../../sites/default/files/pizza-icon.png');
}


Second, we will need a libraries file in the module's root (where pizza.info.yml is located); call it pizza.libraries.yml. Inside this file write something along the lines of this:

toolbar-icon:
  version: 1.0
  css:
    base:
      css/pizza.css: {}


Here, base probably should be changed to theme, since this is an aesthetic to the module.

Third, move into pizza.info.yml, we need to include the library we made into the module. Add this to the bottom:

libraries:
  - pizza/toolbar-icon


Fourth, move into pizza.module, where we're going to add this:

/*
 * Implements hook_toolbar_alter()
 */
function pizza_toolbar_alter(&amp;$items) {
  $items['administration']['#attached']['library'][] = 'pizza/toolbar-icon';
}


This fixed it for me.
Source:
https://www.drupal.org/docs/8/creating-custom-modules/adding-stylesheets-css-and-javascript-js-to-a-drupal-8-module