Add the raspberries, ½ cup black berries, sugar, 3 thyme sprigs, and lemon zest to a heavy saucepan with ⅓ cup water.
Simmer the berry mixture for 10 minutes, stirring occasionally and smashing the berries along the inside of the pot with a spoon.
Strain the berry mixture through a fine mesh strainer over a bowl, pushing on the solids to extract the juices. Chill the berry syrup until ready to use.
Place the remaining blackberries on a plate in a single layer and freeze until solid.
Juice 2 of the blood oranges.
Cut small wedges (to use as garnish) from the remaining orange.
Pour 1 ounce of the chilled berry syrup into a Champagne flute.
Add 1 ounce of blood orange juice.
Top with enough Cava to almost reach the top of the glass.
Drop a frozen blackberry into the mimosa.
Garnish with an orange segment and a small piece of thyme.
Repeat the process for assembling and garnishing the remaining mimosas. Serve immediately.