Line a rimmed baking sheet with parchment paper. In a medium-sized bowl, stir together yogurt, honey, salt, and vanilla until well combined. Transfer one-third to a blender or small food processor and add ½ cup berries and thyme. Blend until smooth.
Spread yogurt mixture on the parchment paper, alternating strips of white and blended berry yogurt. Spread to about ¼-inch thickness. Use a butter knife or small spatula to swirl colors together in a marbled pattern. Sprinkle with nuts and additional berries, if desired.
Freeze until firm, at least 3 hours. Remove from parchment, break into pieces, and serve immediately.