Process flour, oats, butter, and sugar together in a food processor until mixture resembles a coarse meal.
Stir half-and-half and eggs together in a bowl, reserving 1 tablespoon. Add egg mixture to flour mixture in food processor and pulse until it forms a soft dough.
Turn out dough onto lightly floured work surface. Press dough into a circle about 1-inch thick. Gently press blackberries into top of dough; brush with reserved 1 tablespoon egg mixture and sprinkle with thyme. Cut into six wedges and transfer to a baking sheet.
Bake in preheated oven until scones turn golden brown, about 15 minutes.