Preheat the oven to 350 degrees F (175 degrees C). Grease a 9-inch pie dish with 1 teaspoon of olive oil.

Heat 1 tablespoon of olive oil in a skillet over medium heat; cook and stir onions and garlic in the hot oil until garlic becomes fragrant, about 2 to 3 minutes.

Add chopped kale, kosher salt, and black pepper; cook, stirring occasionally, until the kale wilts, which will take about 5 minutes.

Stir in the diced tomatoes and shredded carrots into the kale mixture and cook for 5 more minutes. Once cooked, remove the skillet from the heat.

In a mixing bowl, whisk together the eggs and milk. Stir in the shredded Cheddar cheese, the kale mixture, and chopped parsley into the egg mixture.

Pour the combined mixture into the prepared pie dish.

Bake in the preheated oven until the quiche is set in the middle and a knife inserted in the center comes out clean, which will take about 50 minutes.

Allow it to cool in the pie dish for 2 to 3 minutes before slicing.