Preheat oven to 400F Line a baking sheet with parchment paper.
Put flour, sugar, baking powder, baking soda, and salt in the bowl of a food processor and pulse to combine.
Add the butter pieces and pulse about 30 times to incorporate it, the mixture will be crumbly.
Add the half and half to the processor and process briefly just until the dough comes together. It will be soft and sticky.
Transfer the dough to a bowl and fold in the cherries with clean hands.
Mix just enough to get the cherries evenly distributed. The mixture will be sticky.
Turn the dough out onto a floured surface and knead once or twice, adding a little more flour if necessary to get a soft dough that isn't overly sticky. Pat or roll out to about an inch thick and cut out 6-8 scones with a 2-3 inch biscuit cutter. Note: reform the dough as needed but try not to over work it or the scones can become tough.
Refrigerate scones for about 15 minutes before baking (optional). I think it helps them rise higher. After chilling, arrange the scones onto a parchment lined baking sheet, 2 inches apart.
Bake for about 15-16 minutes, until golden. Cool on a rack before glazing.
To make the glaze, mix the sugar with enough half & half to make it drip off the end of a spoon. Add the extract if desired. Drizzle the cooled scones liberally.