In a small saucepan over medium-high heat, cook the Granulated Sugar (1/2 cup) and Water (2 tablespoon) swirling gently until the sugar dissolves and the mixture turns a pale golden color, 8 to 10 minutes. Lift the pan from the heat and pour in the Banyuls Vinegar (1/2 cup). Return the pan to the heat and cook until the caramel is melted and smooth, 2 to 3 minutes.
Add the Banyuls Fortified Wine (1/4 cup) and Star Anise (1 piece) if using. Simmer until the liquid reduces to thin syrup, about 10 minutes. Keep the gastrique warm over very low heat until ready to use.
Preheat the oven to 450 degrees F (230 degrees C).
Blot the Skin-on Duck Breast (4) dry. Use a sharp knife to score the fat of each in a diamond pattern, taking care not to cut into the meat. Season both sides of each breast generously with Kosher Salt and Freshly Ground Black Pepper (1 dash) then place them skin side down in a large, cold skillet.
Place the skillet over low heat and cook for 3 minutes. Increase the heat to medium, and continue cooking until the duck begins to sizzle. Continue cooking undisturbed until the skin is browned, crisp, and has rendered most of its fat, 6 to 8 minutes.
Spoon off and reserve the fat from the skillet. Flip over the breasts and transfer the skillet to the oven. If you place pan in oven, roast for 2 to 4 minutes or until an instant-read thermometer inserted into the thickest part of the breast registers 125-135 degrees F (51-57 degrees C). Transfer breasts to a plate to rest.
Add the Red Seedless Grapes (1 1/2 cup) to the skillet and toss with the Fresh Thyme (1 teaspoon) and a pinch of salt and pepper. Roast in the oven until the grapes are hot and wrinkled in spots, 8 to 10 minutes.
Transfer breasts to plate, spoon grapes over top. Spoon sauce over top. Top with a small handful of Mâche (1 cup) if using, and serve immediately.