How to Make It
Make the chicken liver mousse: Melt 2 tablespoons butter in a large skillet over medium-high. Add chicken livers and thyme. Cook until livers are browned and medium-rare, 2 to 3 minutes per side. Remove and discard thyme. Transfer livers to a paper towel–lined baking sheet. Let cool completely.
Add shallots, 1 tablespoon kosher salt, and garlic to skillet; cook over medium-high, stirring often, until translucent, about 2 minutes. Add brandy, scraping to loosen browned bits from bottom of skillet. Cook until slightly reduced. Add 1 1/4 cups cream and 1 1/4 teaspoons granulated sugar. Bring to a boil, and cook until slightly reduced, about 5 minutes. Remove from heat, and let cool completely, about 20 minutes.
Transfer cooled cream mixture and cooled chicken livers to a blender. Process until smooth. Pour mixture into a bowl; cover and refrigerate until mixture is chilled and firm, at least 3 hours and up to 3 days.
Make the choux pastry: Combine flour and 2 1/2 tablespoons granulated sugar in a small bowl. Combine 1 cup water, 1/2 cup butter, and 1 teaspoon kosher salt in a medium saucepan. Cook over medium, stirring, until butter is melted. Increase heat to medium-high, and bring to a simmer. Remove from heat, and stir flour mixture into butter mixture with a wooden spoon. Stir until mixture has a paste-like consistency, about 2 minutes. Return to heat over medium, and cook, stirring rapidly, until mixture pulls away from sides of pan and bottom of pan is clean, 1 to 2 minutes. (The dough will be glossy and smooth but not dry.) Immediately transfer dough to a stand mixer fitted with the paddle attachment. Beat on low speed to cool slightly and allow for evaporation, 1 minute. Add eggs, 1 at a time, beating until fully incorporated after each addition. Beat until dough pulls away from the sides of bowl.
Cut parchment or wax paper into 14 (4-inch) squares. Transfer choux pastry to a pastry bag fitted with a large star tip. Pipe dough in a 3-inch round on each parchment square. Freeze until firm, about 30 minutes. Pour canola oil into a Dutch oven to a depth of 3 inches; heat over high to 360°F. Working in batches, place frozen dough rounds on parchment, parchment side up, in hot oil. Fry until golden brown, discarding paper as it detaches, about 5 minutes. Flip and fry 5 minutes. Remove and drain on paper towels. Let cool completely, about 10 minutes.
Make the black honey glaze: Bring honey to a boil in a medium saucepan over high. Cook until honey is mahogany in color and starts to smell slightly burnt, 6 to 8 minutes. Stir in half-and-half, vinegar, powdered sugar, and gelatin. Reduce heat to medium-high, and cook until mixture coats the back of a spoon, about 8 minutes. Remove from heat, and stir in 2 tablespoons cream and luster dust, if using. Using an immersion blender, process until smooth and glossy.
Assemble the Paris-Brests: Transfer mousse to a pastry bag fitted with a large star tip. Using a serrated knife, cut each pastry horizontally into 2 rounds. Pipe mousse in a circle on each bottom pastry half. Dip top pastry halves, top sides down, into honey glaze. Lift out of glaze and allow excess to drip off. Place top pastry round on mousse, and sprinkle with flaky salt.