Preheat oven to 350 degrees F (175 degrees C). Grease a 9-inch pie dish with 1 teaspoon olive oil.
Heat 1 tablespoon olive oil in a skillet over medium heat; cook and stir onion and garlic in the hot oil until garlic is fragrant, 2 to 3 minutes. Add kale, salt, and pepper; cook, stirring occasionally, until kale is wilted, about 5 minutes. Stir tomato and carrots into kale mixture; cook for 5 more minutes. Remove skillet from heat.
Whisk eggs and milk together in a bowl. Stir Cheddar cheese, kale mixture, and parsley into egg mixture; pour into the prepared pie dish.
Bake in the preheated oven until quiche is set in the middle and a knife inserted in the center comes out clean, about 50 minutes. Cool in pie dish for 2 to 3 minutes before slicing.