In a small saucepan over medium-high heat, cook the sugar and water, 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 vinegar. The caramel will bubble vigorously and possibly seize and harden. Return the pan to the heat and cook until the caramel is melted and smooth, 2 to 3 minutes. Add the wine and star anise, if using. Simmer until the liquid reduces to thin syrup, about 10 minutes. Discard the star anise. Keep the gastrique warm over very low heat until ready to use.

Blot the duck breasts 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 salt and pepper, 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.

Heat oven to 350ºF.

Spoon off and reserve the fat from the skillet. Flip over the breasts and transfer the skillet to the oven. Alternatively, flip the breast and cook stovetop until the breast registers 125ºF-130ºF. I prefer finishing the breasts stovetop as I feel I have more control/vision on when the breasts look done, at which point I test with my instant-read thermometer. For me it’s been about 2 minutes on the second side. 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 125ºF-135ºF — Note: I find 125ºF-130ºF to be about right for medium rare. Transfer breasts to a plate to rest.
Add the grapes to the skillet and toss with the thyme 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, if using, and serve immediately.