Note: The marinated grapes require overnight resting. Marinate the grapes: In a 1-quart jar, combine the white wine, vinegar, sugar, and salt. Shake or stir well, then add the grapes. Cover and refrigerate overnight.
An hour before serving, make the blini batter: In a large bowl, whisk the flour, 2 1⁄3 cups water, the oil, yeast, sugar, salt, and eggs until well combined. Let stand at room temperature until risen and bubbly, about 1 hour.
Meanwhile, make the pâté: In a large skillet over medium-high heat, melt 4 tablespoons butter. Once hot, add the chicken livers and season with 3⁄4 teaspoon salt; cook, turning occasionally, until just heated through, 10–12 minutes total. Remove to a food processor. In the same skillet over medium-high heat, melt the remaining 2 tablespoons butter. Add the onion, carrot, and remaining 1⁄2 teaspoon salt and cook, stirring occasionally, until lightly browned, about 8 minutes.
In a food processor or blender, put the vegetable mixture and the prepared liver. Season generously with salt and add the red wine; blend or process until very smooth. Taste and adjust the seasoning as needed. Press the mixture through a fine-mesh sieve if a smoother consistency is desired. Pâté can be served warm or at room temperature and can be made up to 2 days ahead.
When ready to serve, place a paper towel-lined platter or a rack set over a baking sheet next to the stove. Whisk the blini batter briefly. In a medium nonstick skillet, melt 1⁄2 tablespoon butter over medium-high heat. Drop scoops of the batter by the tablespoon into the pan, leaving room between the blini; cook, turning once, until golden, about 3 minutes per side. Transfer to the rack or paper towel-lined platter and repeat with the remaining butter and batter (work quickly to avoid burning the butter).
To serve, spread the pâté on blini and top with grapes.