Preheat oven to 325ºF (160ºC).
To roast vegetables, line a pan with aluminum foil and spread tomatoes and corn (you can do it separately if you want). Don't mix them, keep them separated even if using the same pan.
Season with salt, pepper and a drizzle of olive oil.
Roast until tomatoes begin to shrivel and caramelize and corn is charred in some places, about 30 minutes. The time might vary, depending on your oven.
Heat some 2 tablespoons of olive oil and the butter in a large skillet.
Add the chopped shallot, corn and thyme. Cook for a minute or two over medium heat, until the shallot is softened.
Add the cream and parmesan. Season with salt and pepper and stir to combine. Remove from the stove and reserve.
Cook the pasta in abundant salted water according to the manufacturer's directions. Drain and transfer to the skillet. Reserve 2-3 tablespoons of the pasta water.
Add the tomatoes, basil and half the chives to the pasta. Stir everything together for a minute or two until the pasta is coated. Add some of the reserved pasta water to bind the mixture.
Divide between two serving plates and serve topped with a drizzle of olive oil and more minced chives.