Fill a large saucepan with water and bring to a boil.
Meanwhile, line a large plate with paper towel. Heat a large skillet on medium-high and mist with cooking spray. Add sausage and cook, breaking up meat with a wooden spoon, until no longer pink, 7 to 9 minutes. With a slotted spatula, transfer sausage to plate and cover with 2 more layers of paper towel, pressing lightly to absorb excess fat; set aside.
Preheat oven to 350°F. To boiling water, add kale and cook until just tender, 1 to 2 minutes. With a slotted spoon, transfer kale to a colander and set aside to drain. Return saucepan to a boil, add pasta and cook until just barely al dente (undercooked by about 1 minute). Meanwhile, line a large bowl with paper towel and transfer kale to bowl to absorb any remaining liquid. Drain pasta and set aside.
In a separate large saucepan, melt butter on medium-low. Add garlic and cook, stirring frequently, for 1 minute, until fragrant. Gradually whisk in flour and cook, whisking constantly, for 2 minutes. Slowly whisk in 1 cup milk. Increase heat to medium-high and simmer, whisking frequently, until slightly thickened, 2 to 3 minutes. Slowly whisk in remaining 3 cups milk. Bring to a simmer and cook, whisking frequently, for 5 minutes. Reduce heat to low, add mozzarella and whisk until melted and mixture thickens, 1 to 2 minutes. Turn off heat.
To milk mixture, add tomatoes, salt, pepper, sausage, pasta and kale and stir to combine. Transfer to a 9 x 13-inch baking dish and top with Parmigiano-Reggiano. Bake until cheese melts and sauce bubbles, 15 to 18 minutes. Let rest for 10 minutes before serving.
(MAKE AHEAD: Prepare up to 24 hours ahead, but don’t bake. Cool to room temperature, then cover and refrigerate. To serve, bake covered for 20 minutes, then uncover and bake for 15 to 20 minutes, until sauce is bubbly. You can bake this dish at home then reheat, covered, at your host’s place in a 350°F oven just until warmed through.).