Preheat the oven to 375 °F. Lightly grease a 2-quart baking pan.

Put a skillet over medium heat and when it is hot, add the oil. Add the onion and cook about 10 minutes, until tender.

While the onion is cooking, put eggs and milk in a bowl and mix until combined.

Add the cheese, bread, kale, squash, and salt and mix well. Let the mixture stand at least 15 minutes until the bread absorbs most of the milk.

When the onion has finished cooking, add it to the bread mixture and mix well.

Pour the mixture into the prepared pan and transfer to the oven.