Add the chopped tomatoes, sliced onion, crushed garlic, butter, balsamic vinegar, sugar, and salt straight into your slow cooker. Set it to HIGH for a minimum of 3 hours, or LOW if you'd rather let it go for 5 hours or longer. I like to lift the lid once, about halfway through, just to give everything a quick stir once the butter has melted in.
You'll know it's ready to move forward once the tomatoes have completely broken down and the onion has turned soft and translucent. Every slow cooker runs a little differently, so keep an eye on it the first time you make this. If you're not quite ready to eat, simply switch it over to LOW and let it hold until you are.
Once the tomato mixture is fully cooked down, it's time to blend. I always reach for my immersion blender since I can blend everything right in the pot without any mess. If you're using a countertop blender instead, let the mixture cool slightly first before transferring it in batches.
Once your soup is silky smooth, stir in the parmesan and double cream until fully combined. Add the fresh basil leaves and pulse briefly, just enough to distribute the flavor while leaving small green flecks visible throughout the soup
If you blended everything with an immersion blender, you can serve right away or switch the slow cooker to the KEEP WARM setting until you're ready to eat. If you used a countertop blender and the soup cooled down, just return it to the crockpot for another 15 to 20 minutes to warm back through before serving.