Although it is not documented, the icons used in the touch bar for Favicon are the same as the Safari pinned tab icon.
First of all, you need a black and white SVG of your logo in a 45×45 pixel layout uploaded to the root directory of your site.
Then, in the header of your website you need to add the following code:
<link rel="mask-icon" href="https://pingie.com/PingieLogoNEWSVG.svg" color="#5bbad5">
Replace the Pingie URL in the HREF with the link to the SVG you uploaded to your website.