Cook the potatoes. Place the potatoes in a large pot and cover with salted water. Bring to a boil and cook until the potatoes are fork tender. This usually takes 15 to 20 minutes.
Make the tomato butter. While the potatoes cook, heat a drizzle of olive oil in a skillet. Add the sliced shallot and sauté for about 3-5 minutes until soft. Add the tomatoes and cook until they start to burst, about 5 to 8 minutes. Press gently with a spatula to release the tomato juices. Add the butter and stir until melted. Season with salt and remove from the heat.
Mash the potatoes. Drain the potatoes and return to the pot. Mash with a potato masher. Leaving some skin adds good texture.
Combine. Pour the tomato butter mixture into the mashed potatoes. Use an immersion blender to lightly blend everything together. Stop before the potatoes are completely smooth. Some texture is ideal.
Add in the cheese. Fold in the Gruyère. Add half first, stir to combine, then add the rest. Stir until the cheese is fully stirred in.