Grind fennel seeds and peppercorns into a coarse powder, using a spice grinder or a mortar and pestle. Transfer to a bowl and combine with rosemary, crushed red pepper flakes, thyme, brown sugar, and 1 tbsp of olive oil.
Coat pork in 1 tbsp of oil and 1½ tablespoons of salt. Next, using your hands, evenly coat the pork in the spice mixture. Let stand for 30 minutes - 1 hour or until it reaches room temperature.
Preheat oven to 300℉. When ready, heat a cast iron skillet over medium high heat until hot. Place pork into hot pan and sear until golden brown, about 1 - 2 minutes. Flip and repeat until all sides are brown. Remove from the skillet and set aside.
Add the onion and carrot to the pan (add a little olive oil if dry). Stir frequently and cook until softened slightly, about 3 minutes. Add garlic. Cook another minute. Add the broth, using a wooden spoon to scrape off the browned bits. Add the thyme sprig and season with salt and pepper.
Add pork shoulder to the center of the skillet. Place in your preheated oven, uncovered. Cook until the internal temp reaches 155℉, about 1½ hours. (It will reach 165℉ when resting).
Remove from the oven and allow to completely cool, about 1 hour. Strain the liquids, discarding the solids, into a resealable container. Once cool, tightly wrap the pork in plastic wrap. Refrigerate both to use the next day. (You don't have to do this a day ahead. But we found wrapping it and refrigerating it overnight locks in the juices and makes it easier to thinly cut, because it's cold).