Remove the outer layer of your brussel sprouts looking for dark, discolored or broken leaves. Trim the end of the sprout, and then slice as shown above, slicing around the core. Discard the core. This takes a few minutes. Alternatively, don’t worry about the core, use a food processor fitted with a slicing blade and slice the whole batch in under a minute.
On a medium-high burner, heat 1/4 cup olive oil in a large Dutch oven or other heavy-bottomed pan. Add shallots and garlic and saute for 10 minutes, stirring until they begin to caramelize. Add shaved sprouts and cook for another 10 minutes, until the sprouts begin to caramelize. Season with salt and pepper.
Remove from heat and stir in parmigiano, pumpkin seeds, lemon juice, lemon zest, pomegranate seeds and mint. Taste and adjust salt, pepper and lemon juice. This dish loves lemon juice. Serve hot or at room temperature.