The tartness and texture of the blackberry conserva is a nice contrast to the rich, silky pâté. Spoon leftover blackberry conserva on pancakes, waffles, or ice cream.
Stir together livers and 1/4 teaspoon salt in a medium bowl; set aside.
Melt butter in a large skillet over medium. Add onion, and cook, stirring occasionally, until softened and translucent but not browned, about 10 minutes. Transfer onion mixture to bowl of a food processor. Add livers to skillet, and cook over medium until bright pink all the way through, 1 minute to 1 minute and 30 seconds per side. Add sage, and cook until fragrant, about 15 seconds. Remove skillet from heat; stir in Champagne. Let stand 5 minutes.
Transfer liver mixture to food processor with onion mixture. Process until smooth and silky, about 2 minutes, stopping to scrape down sides as needed. Add remaining 1 tablespoon salt. Process until incorporated, about 10 seconds. Using a rubber spatula, press liver mixture through a fine wire-mesh strainer into a bowl; discard solids. Transfer pureed liver mixture to a large (20- to 24-ounce) terrine mold or ramekin or 2 smaller (10- to 12-ounce) ramekins. Smooth top, and tap ramekin on counter a few times to remove any air bubbles. Cover pâté with melted duck fat (about 1/4 inch thick), and chill, uncovered, until set, at least 6 hours or up to 2 days.
Combine blackberries, sugar, and 2 tablespoons water in a large saucepan. Wrap thyme and cloves in cheese-cloth, and tie ends to make a sachet. Add sachet to blackberry mixture. Bring to a boil over medium-high, stirring often. Reduce heat to medium-low; cook, stirring often, until blackberries break down and mixture has thickened and reduced to about 2 cups, 30 to 40 minutes.
Remove from heat. Remove and discard sachet. Stir in salt and pepper. Transfer blackberry mixture to a shallow bowl, and spread out to cool. Immediately transfer to refrigerator, and chill, uncovered, 2 hours to preserve vibrant color. Transfer blackberry conserva to an airtight container, and chill at least 2 hours or up to 3 days.
Remove pâté from refrigerator 30 minutes before serving. Serve pâté with blackberry conserva, grilled crusty bread, and whole-grain mustard.
Duck liver pâté can be made up to 2 days in advance; blackberry conserva can be made 3 days in advance.
Robust but elegant rosé Champagne.