Preheat your oven's broiler to high and line a rimmed baking sheet with parchment paper. Spread the cherry or grape tomatoes in a single layer and broil for 6 to 8 minutes, watching closely until the skins blister and start to turn dark brown.

In a food processor, combine the cooled tomatoes with the fresh thyme leaves, kosher salt, and freshly ground black pepper. Pulse the mixture until it is finely chopped but still has some texture, about 10 pulses.

Add the cubed, softened unsalted butter to the tomato mixture in the food processor and pulse until well combined, about 30 seconds.

Transfer your Tomato Butter to a covered dish or roll it into a log shape for easy storage. Refrigerate for at least 2 hours before using.