Remove breasts from partridges, leaving bones attached to carcasses. Discard skin; chill breasts. Chop carcasses. Heat oil in an 8-qt. saucepan over medium-high. Cook carcasses until browned, 6–8 minutes. Add carrot; cook until soft, 5–7 minutes. Add stock, dried and fresh blueberries, thyme, and bay leaves; boil. Reduce heat to medium-high; cook, skimming fat as needed, until reduced by half, about 1 hour. Strain stock and return to pan; simmer until reduced to 1 cup, 20–22 minutes. Simmer cream in a 1-qt. saucepan until reduced by half, 8–10 minutes; whisk into stock. Remove from heat; whisk in 2 tbsp. butter, salt, and pepper. Keep sauce warm.
Melt remaining butter in a 12" skillet over medium-high. Season reserved breasts with salt and pepper. Cook, flipping once, until browned and an instant-read thermometer inserted into the thickest part of the breast reads 130°, 3–4 minutes. Slice breasts; spoon sauce over the top.
