Pre-heat the oven to 400ºF.
Make soup: Cut garlic and shallots into large chunks. In a large pot heat 1 tablespoon of olive oil, garlic, and shallots and sauté until lightly browned. (Be careful not to burn the garlic, this step should only take between 5 and 7 minutes). Separate the stem from the tops of the shiitakes. Thinly slice the caps and set both aside.
Add the stems to the garlic and shallot mixture. Add the teaspoon of thyme. When the veggies start to stick on the bottom of the pot, add mirin to deglaze. Stir well. Add the broccoli and 6 cups of water. Add a good pinch of salt and pepper. Cover and reduce heat. Simmer until the broccoli stalks are completely soft. Then using an immersion blender, blend until smooth.
Make the shitake chips: On a tray lined with parchment paper, place sliced shitake caps, a tablespoon of olive oil, and pinch of salt and pepper. Using your hands, mix until combined. Bake for 10 minutes or until soft, then the pan and leave in the hot oven for another 5 minutes.
To serve: ladle soup into bowls and top with chips.