To make the sauce, put the cherries in a pan with the thyme and sugar and add 1 tablespoon water (but not if using frozen cherries).
Bring to a simmer and cook very gently for a few minutes until the fruit is soft but not collapsed. Taste and add a little more sugar if needed, but keep it quite tart. Set aside.
Toss the pigeon breasts with the olive oil, thyme leaves and some salt and pepper. Heat a frying pan over a medium-high heat and add the pigeon breasts.
Cook for 1 minute, turn the breasts over and cook for another minute, then turn again. Toss them in the pan for a couple more minutes.
Slice one pigeon breast in half. If it is nicely pink in the middle, remove all the breasts from the pan and set aside to rest on a warm plate.
If it still looks very raw, cook for another minute or two. Once the pigeon breasts are out of the pan and resting pour the wine into the hot pan to deglaze it, letting it bubble briefly while you scrape up any caramelised bits with a spoon.
Add the cherry sauce and stir briefly to combine with the pan juices, then taste and add more salt and pepper as needed.
Cut each pigeon breast on the diagonal into 4 or 5 slices. Arrange on warm plates. Add the warm cherry sauce, along with any resting juices and it’s ready to serve.
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