Bring a large pot of salted water to a boil over medium-high heat. Add the cauliflower florets and boil until fork tender, 7 to 9 minutes. Drain the water and set aside to cool.
Heat a large pot or dutch oven over medium heat. Add the butter and oil. Once melted, add the shallot. Cook, stirring often, until translucent, 1 to 2 minutes. Add the garlic and continue cooking and stirring until fragrant, about 1 minute.
Add the tapioca flour, salt, and ground black pepper. Stir until a paste-like mixture forms. Pour in the broth and stir to release any flour stuck to the pot.
Increase the heat to high and bring to a boil. Once boiling, stir in the broccoli, cover the pot, and cook until soft, 12 to 15 minutes.
While the broccoli cooks, add the cooled cauliflower and cottage cheese to a blender. Blend on high speed until completely smooth, 45 to 60 seconds. If needed, add water or broth 1 tablespoon at a time to thin and blend until smooth.
After the broccoli has cooked, remove the pot from the heat and remove the lid. Cool the soup for about 5 minutes. Once slightly cooled, stir in the cauliflower and cottage cheese mixture and the shredded cheese, and mix to combine, allowing the cheese to melt.
To serve divide evenly among four bowls.
