In a food processor, pulse the flour and 1 tablespoon kosher salt to combine. Add the butter and pulse until it forms pea-size crumbles. Add the egg yolks and continue pulsing until dough just comes together. Transfer the dough to a clean work surface and knead until smooth, about 2 minutes. Cut off one-third of the dough and shape it into a disk. Shape the remaining two-thirds into another disk and wrap both disks separately in plastic wrap. Refrigerate both doughs for 30 minutes or until ready to use.
Using a spice grinder, grind the thyme, anise seeds, and porcini mushroom into a fine powder and transfer to a large bowl. Stir in the remaining 2 tablespoons kosher salt along with the pepper, fennel pollen, and pickling salt. Add the duck, chicken, and livers and toss until evenly coated in the spices. Transfer the bowl to the freezer and chill until the meat is firm but not frozen, about 30 minutes. Set a meat grinder to the large dice setting and pass the meat through it and into a bowl. Pass half the ground meat through the grinder a second time. Stir the ground meats together until evenly combined, then cover the bowl with plastic wrap and refrigerate at least 8 hours or overnight.
In a 10-inch skillet, heat the olive oil over medium-high. Add the onion and cook, stirring, until soft, about 3 minutes. Add the trumpet mushrooms and cook, stirring, until tender, another 5 minutes. Remove the skillet from the heat and let cool completely. Scrape the onions and mushrooms into the ground meat and stir to combine.
Heat the oven to 350°. Using a rolling pin, roll the larger dough disk into a 12-by-18-inch rectangle, about 1⁄6-inch thick. Place a 1 1⁄2-quart rectangular terrine mold in the center of the rectangle and lightly score the dough around the base of the mold, making sure not to cut through the dough. Roll the remaining dough disk into a 18-by-8-inch rectangle, about 1⁄6-inch thick, and trim to a 12-by-4-inch rectangle, reserving the scraps. Using a 1 3⁄4-inch round cutter, cut a circle out of the dough scraps, then, using a 1-inch round cutter, cut out the center of the dough circle to make a ring of dough.
Place the larger rectangle of dough into the terrine mold, using the score marks as a guide, and let the excess dough drape over the edges. Firmly press the dough into each corner of the mold, then fill with the ground meat, making sure to pat it down as you add to prevent air pockets. Fold the excess dough back over the meat, making sure it is completely covered. Brush the dough with the beaten egg then place the 12-by-4-inch dough rectangle on top, pressing firmly to seal and tucking the dough edges into the mold.
Position a 1⁄2-inch round cutter at the one-third mark of the top of the dough and cut a vent in the dough. Brush the dough with the beaten egg and place the reserved dough ring over the vent. Cut the remaining dough scraps into decorative leaves and circles and place them on top of the terrine. Brush the decorations with the beaten egg.
Roll a 2-inch square of foil into a cylinder and fit it inside the vent, like a chimney (this ensures the vent keeps its shape and properly lets out steam). Bake the terrine until the dough is golden brown and the center of the pâté reaches 165°, about 1 hour. Let the terrine cool for 1 hour, then transfer to the refrigerator.
In a small bowl, whisk the gelatin with 1⁄4 cup chicken stock and let stand for 5 minutes. In a small saucepan, bring the remaining stock to a simmer, then remove from the heat and whisk in the gelatin until dissolved. Remove the pan from the heat and let the stock cool to room temperature. Using the foil chimney as a funnel, slowly pour the stock into the vent of the terrine and chill the pâté until the gelatin is set, at least 8 hours or overnight.
To serve, remove and discard the foil chimney, then carefully invert the chilled pâté en croûte and remove it from the mold. Invert it right side up, cut into 1-inch thick slabs, and serve the pâté with mostarda.