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Bring 4 quarts of water to a boil in large pot and cook ziti according to box directions to only al dente and set aside.
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At the same time, lightly steam broccoli florets on to of pan in steamer and set aside.
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Meanwhile, melt butter in large nonstick skillet over medium heat. When just coming to foaming, add garlic paste and cook until fragrant, about 1 minute. Add flour and whisk until beginning to turn golden. Add vermouth and cook until evaporated. Stir broth into pan and simmer sauce until slightly thickened, about 2 minutes. Add cheddar and stir constantly while returning to a boil. Fold in ziti cook for one minute additional and remove from heat. Add broccoli and chicken and fold until sauce covers uniformly.
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Plate and top with one or two strips of parmesan (optional) and serve.