Heat the oil in a 7- to 8-quart enameled Dutch oven over medium-high heat. Season the chicken generously with salt. Working in batches so as not to crowd the pan, sear the chicken, turning once, until golden-brown, 10 to 12 minutes per batch. Transfer the chicken to a large bowl.
When all the chicken is browned, pour off all but 2 Tbs. of fat. Add the grapes and cook until just tender, about 3 minutes. With a slotted spoon, transfer the grapes to a small bowl. Reduce the heat to medium and add the onions and a pinch of salt to the pan. Cook, stirring frequently, until tender, about 7 minutes. Add the garlic and cook, stirring, to soften, 1 to 2 minutes. Pour in the wine and simmer, stirring up the browned bits on the bottom of the pan, until the wine reduces by almost half, about 5 minutes.
Return the chicken to the pan, along with any accumulated juices. Add the broth and sprinkle with thyme. Bring to a simmer, reduce the heat to medium low, cover, and cook until the chicken is very tender, about 25 minutes. Transfer the chicken to a bowl.
Raise the heat to medium high and boil the liquid until reduced to about 2-1/2 cups, about 12 minutes. In a small bowl, whisk the cream with the cornstarch, then whisk the cream mixture into the sauce. Cook, whisking constantly, until the sauce simmers and thickens to the consistency of heavy cream, 1 to 2 minutes. Season to taste with salt and pepper. Return the chicken to the pan. Simmer gently over medium-low heat until the chicken is heated through. Stir in the reserved grapes. The chicken skin will be soft—remove it prior to serving, if desired.
This dish can be prepared 2 days ahead; cool, cover, and refrigerate the chicken and grapes separately.