Peel the zucchini lengthwise into long strips and arrange the ribbons on a paper towel-lined baking sheet to dry.
Heat olive oil in a large skillet over medium-high heat. Season chicken with thyme, salt, and pepper, then cook about 4 minutes per side until cooked through.
Remove the chicken, cut it into strips, and cover to keep warm.
In the same skillet, heat butter and olive oil over medium heat. Cook garlic and tomato paste for 30 seconds, then add diced tomatoes. Bring to a boil, reduce heat, and simmer 10 minutes. Add vodka and simmer 8 to 10 minutes.
Stir in milk and simmer 10 minutes more. Add Parmesan and stir until incorporated.
Return chicken and zucchini to the sauce and cook 1 minute, until heated through. Remove from heat and serve.