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Score the skin of each duck breast diagonally with a sharp knife, taking care not to pierce meat, and season both sides with salt and pepper.
In a large, heavy frying pan over medium-high heat, melt 2 tablespoons butter until it foams. When foaming subsides, add duck breasts, skin side down, and sear until skin is browned and fat begins to render, about 8 minutes. Turn over duck and reduce heat to medium. Cook until duck registers 130° on an instant-read thermometer for medium-rare, 8-10 minutes. Transfer duck to a rack set over a rimmed baking sheet. Tent loosely with foil.
Pour off all but 2 tablespoons of fat from pan and return to medium heat. Add shallot and cook, stirring frequently, until softened, 3-4 minutes. Add broth and port and cook, scraping pan bottom to release browned bits. Add cherries and 1 tablespoon thyme and cook until thickened, 7-8 minutes. Whisk in remaining 1 tablespoon of butter. Taste and season with salt and pepper. Keep warm over low heat.
To assemble, using a sharp slicing knife, slice breasts into 1/4” thick slices on the diagonal. Divide slices evenly between 4 warmed serving plates, overlapping the slices slightly. Ladle about 2-3 ounces of sauce over the duck slices. Garnish with thyme sprigs.