Preheat oven to 400°F (200°C). Line a baking sheet with parchment paper.
In a mixing bowl, toss cherry tomatoes with 1 tablespoon olive oil, minced garlic, fresh thyme (or oregano), salt, and pepper until evenly coated.
Spread the tomatoes on the prepared baking sheet in a single layer and roast for 15-20 minutes, until soft, blistered, and just starting to burst.
Meanwhile, crumble the block feta cheese into a food processor.
Add Greek yogurt, 2 tablespoons olive oil, fresh lemon juice, lemon zest, and black pepper to the processor.
Blend until smooth and creamy, scraping down the sides as needed for an even consistency.
Taste and adjust seasoning with salt, pepper, or extra lemon juice as preferred.
Transfer the whipped feta dip to a serving bowl.
Top the dip with warm roasted tomatoes and their juices.
Drizzle with additional olive oil and garnish with fresh basil leaves if desired.
Serve immediately with toasted baguette slices, pita chips, or fresh vegetable sticks.