Heat a large, deep skillet over medium heat. Add the chorizo and sauté for 2 minutes over medium-high heat until it starts to brown and release some of its oils. Remove the chorizo with a slotted spoon to a plate. Add 3 tablespoons of the oil to the pan with any oil left from sautéing the chorizo and heat over medium heat. Add the breadcrumbs, season with salt and pepper, and toast, stirring frequently, until the breadcrumbs are light golden brown, about 4 minutes. Add half of the garlic and cook and stir for one more minute, until you can smell the garlic. Use a slotted spoon to transfer the toasted crumbs to a paper towel lined plate. Wipe out the skillet and return to the heat.
Add the remaining 2 tablespoons oil to the pan, and heat over medium high heat. Add the Brussels sprouts and sauté until the Brussels sprouts are beginning to become tender, and getting browned in spots, about 6 minutes. Add the remaining half of the garlic and sauté for another 30 seconds, until you can smell the garlic. Add the broth, cover the pan, and cook for another 4 minutes until the Brussels sprouts are tender (but not soft!). Uncover the pan, sauté for one more minute until most of the liquid is evaporated, stir in the chorizo, then turn into a serving dish.
Sprinkle the breadcrumbs on top and serve hot or warm.