Preheat the oven to 400 degrees. Line a rimmed baking sheet with parchment paper or foil for easy cleanup.
In a large bowl, combine eggplant, red bell pepper, onion, garlic, and thyme. Drizzle with olive oil, 1 teaspoon salt and 1/2 teaspoon pepper and toss to combine.
Spread on prepared baking sheet and roast until the vegetables are soft and lightly browned, about 45 minutes, stirring halfway through roasting time. Cool 5 minutes.
Transfer to a food processor and add tomato paste. Pulse until blended, 3 or 4 times. Transfer to a bowl and stir in lemon juice, red wine vinegar, sugar, and red pepper flakes. Season to taste with salt and pepper. Serve warm or at room temperature.