Drain and dry the feta: Remove the feta from its brine and pat it dry with a paper towel. This keeps the dip from turning watery once blended.
Add ingredients to processor: Place the feta, pumpkin puree, olive oil, honey, and pumpkin pie spice in the bowl of a food processor fitted with the blade attachment.
Blend until smooth: Process the mixture for 1-2 minutes, stopping once to scrape down the sides, until the dip turns pale orange and fluffy with no visible feta chunks.
Taste and season: Taste the dip and add the salt, or a touch more honey if you prefer it sweeter. Blend again for a few seconds to combine.
Chill briefly: Transfer the dip to a serving bowl and refrigerate for about 15 minutes to firm up slightly and let the spices settle.
Garnish and serve: Swirl the dip with the back of a spoon, drizzle with olive oil, and top with toasted pepitas and fresh thyme or sage before serving.