Bring a large pot of salted water to a boil. Add the pasta and cook as the label directs. Drain, rinse under cold water and return to the pot. Toss with 1/4 cup milk. Top with the cheese and chicken (do not stir).
Meanwhile, melt 1 tablespoon butter in a large skillet over medium-high heat. Add the scallions, chili powder, 1 teaspoon salt and a few grinds of pepper. Cook, stirring, 2 minutes. Add half of the pico de gallo. Cook, stirring, until the scallions soften, 2 to 3 minutes.
Reduce the heat to medium. Add the remaining 2 tablespoons butter and swirl to melt. Sprinkle in the flour; cook, stirring, until a paste forms, about 1 minute. Gradually stir in the broth and the remaining 1 cup milk. Bring to a simmer; cook, stirring occasionally, until thick enough to coat the back of a spoon, about 6 minutes.
Add the sauce to the pot with the pasta. Cook over medium heat, stirring, until the cheese melts, about 3 minutes; season with salt and pepper. Top with the remaining 1/2 cup pico de gallo.