Preheat the oven to 250°F (120°C).
Chop up your pieces of white chocolate until they look splintered, even if you're using white chocolate chips. You want the surface area of the chocolate slivers to be as small as possible, so that each one is evenly exposed to the heat.
Cover your baking sheet with the (cleaned) silicone mat, and scrape on your chopped chocolate, covering the surface evenly, with a single layer.
Put the chocolate-covered tray in the oven and set the timer for 10 minutes.
When the timer goes off, take out the chocolate and use your spatula to move around the melted chocolate, ensuring that it browns evenly. Return it to the oven for another 10 minutes, repeating this process one more time (30 minutes total), and then every 5 minutes afterwards until the chocolate is smooth and deep brown. It should be roughly the color of peanut butter, according to my boss in Guatemala (where I learned this technique many years ago).
Make sure you don't get any water on the chocolate at any time, as that can cause chocolate to seize up and become unworkable, not to mention stop caramelizing. If you ever pull out the tray and see oil separating from the chocolate, never fear; that's cocoa butter separating from the solids in the chocolate, and it will recombine with a few pulses from a food processor or a couple minutes with a hand mixer (AFTER caramelization is done, so don't mess with it yet).
Once the chocolate is sufficiently browned, pull it out of the oven and let it cool. If it's separated, let it cool for 2 minutes before scraping it into a bowl to recombine at a high speed. Then store in an air-tight container for up to a month, no refrigeration required.