Preheat the oven to 425 degrees. Place a sheet of parchment paper on a baking sheet.
Drop small piles (about 2 tablespoons worth) of asiago cheese on the parchment paper 2 inches apart. Stick the pan in the oven and bake for 4 to 5 minutes, or until the cheese is melty and crisp. Remove and let cool for 10 minutes before removing the crisps from the pan.
Toss the quinoa with olive oil. Spread it out on a baking sheet. Place the sheet in the oven and roast the quinoa for 20 to 25 minutes, tossing it every 6 to 8 minutes. Remove the pan and let it cool.
Make the soup while the quinoa is in the oven. Heat a saucepan over medium-low heat and add the butter. Add in the shallots and garlic with a pinch of salt and pepper, cooking until soft and slightly golden and stirring occasionally, about 5 to 6 minutes. Stir in the tomato paste and cook for another 5 minutes. Add in the tomatoes, sherry, chicken stock, brown sugar, basil, another pinch of salt and pepper. Bring the mixture to a boil. Reduce to a simmer and cook for 15 to 20 minutes.
Transfer the soup to a high-powered blender and carefully blend until pureed. Transfer back to the pot over low heat. Right before serving, stir in the heavy cream. Taste and season additionally with salt and pepper if needed – it might, depending on your tomatoes.
Serve the soup in bowls with a scoop of toasty quinoa, a few cheese crisps and fresh basil. Enjoy!