Magento 2.1 Widgets – Add Classes and Icons to Homepage Widget

I am working on a Magento 2.1.7 shop. To achieve this, I have created a child-theme of Magento Blank.

On the home page, I have inserted a "New Products" widget via the Magento admin.

I am using Font Awesome and Bootstrap in my custom theme and I want to take advantage of those, by adding classes and icons to the product boxes in the image below.

I want those "Add to Cart" to have this code:

<button type="submit" title="Add to Cart" class="action tocart btn">
 <span class="hidden">Add to Cart</span>
 <i class="fa fa-shopping-cart" aria-hidden="true"></i>
</button>

The problem is the HTML editor in the Magento admin does not offer this possibility and I can't find the .phtml template responsible for rendering the widget.

1. What is a viable alternative to what I did so far?
2. Is there an alternative to inserting the "New Products" widget via the Magento admin, that would allow me to edit the widget's HTML as detailed as I wish/need? In other words: can I insert the block via some template?

Thank you!