Make the corn stock: Slice off the kernels from the corncobs, reserving the kernels for later. Place the cobs in a medium saucepan with the remaining stock ingredients. Bring to a boil over high heat, then reduce the heat to medium and simmer until reduced to 1½ cups, 1 hour. Strain, discarding the solids.

Make the pasta: Bring a large pot of salted water to a boil. Add the pasta and cook, stirring occasionally, until very al dente, 6 minutes, then drain.

Meanwhile, in a large, deep skillet, toast the pepper over medium heat until fragrant, 1 to 2 minutes. Add the corn stock, vegan butter and corn kernels, and bring to a simmer.

Add the pasta to the pan and cook, stirring constantly, until the pasta is fully cooked and the sauce has reduced slightly, 3 to 4 minutes. Add the cheeses and continue to cook, stirring constantly, until the cheeses have melted and the pasta is fully coated in the sauce, 1 minute more. Divide between bowls and serve immediately.