Cook the pasta in a large pot of generously salted boiling water until just al dente according to package directions; drain and set aside.
While the pasta cooks, heat 1 tablespoon olive oil in a large skillet over medium-high heat. Add the chopped chicken, season with 1/2 teaspoon salt and a pinch (about 1/8 teaspoon) of pepper, and cook until the exterior is no longer pink.
Add the chopped zucchini and corn to the skillet and cook 1–2 minutes more, until the chicken is cooked through and the zucchini is crisp-tender; transfer the mixture to a plate and wipe out the skillet.
In the now-empty skillet, melt 2 tablespoons butter with 2 tablespoons olive oil over medium heat. Whisk in 1/4 cup flour and cook, stirring, for about 2 minutes to form a roux.
Reduce heat to low and gradually whisk in 1 1/2 cups milk, 1 cup salsa verde, the 12-ounce can of evaporated milk, 1/2 teaspoon salt, and 1/2 teaspoon ground cumin until smooth.
Bring the sauce to a gentle boil while whisking constantly, then reduce heat to medium and simmer, whisking occasionally, until it thickens but is not overly stiff.
Remove the sauce from heat and stir in 1 cup freshly grated Pepper Jack until melted, then add 1 cup freshly grated sharp cheddar and stir until fully melted. Taste and add ancho chili powder if desired.
Add the cooked pasta to the sauce and toss until evenly coated. Return the chicken, zucchini, and corn to the skillet and stir until everything is combined. Season with additional salt, pepper, and hot sauce to taste.