Heat a 4-to 6-quart Dutch oven or soup pot over medium heat. Add the bacon and cook, stirring occasionally, until crispy, 8 to 10 minutes. Transfer the bacon to a paper towel-lined plate and drain off all but 2 tablespoons of bacon grease.
Increase the heat to medium-high. Add the olive oil, onion, carrot, celery, and garlic to the bacon grease. Cook until the vegetables are softened, about 5 minutes.
Add the thyme, rosemary, red-pepper flakes, and flour. Stir to coat everything in flour, then pour in the broth. Simmer, stirring occasionally, until the vegetables are tender, and broth has thickened slightly, 15 to 20 minutes.
Stir in the gnocchi, milk, cheddar, Parmesan, and kale. Stir until soup is smooth and cheese is melted. Stir in most of the bacon, reserving some for garnish. The soup may simmer gently, but do not boil.
Season to taste with salt and serve. Garnish with extra bacon and herbs if desired.