Heat the canola oil in a large nonstick saute pan over medium-high heat until it begins to shimmer. Add the sausage and chile flakes and cook, stirring occasionally, until golden brown and cooked through, about 5 minutes.
Add the kale to the pan, sprinkle with salt, and cook, tossing, until the kale is bright green and just starting to soften, about 5 minutes. Stir in the pasta and tomatoes.
Whisk the eggs until fluffy and season with pepper. Stir the eggs into the pan, season with salt, and cook, stirring, until the eggs coat the pasta and some curds form, about 3 minutes. After the first minute or so, stir in the Parmesan and basil. Taste, and season as necessary. Transfer to a large bowl and garnish with additional cheese and basil. Serve immediately.