Place chilies in a large heat-proof bowl. Cover with boiled water and let sit for at least 30 minutes or overnight.
Meanwhile, add caraway seeds, coriander seeds, and cumin seeds to a dry skillet. Toast over the lowest possible heat until fragrant, about 5 minutes. Shake frequently to prevent scorching. Remove from heat.
Transfer toasted spices to a small grinder or mortar and pestle. Process until finely ground.
Drain the chillies well. Remove the stem and seeds from each chili (wearing gloves is recommended) and transfer to a food processor.
To the food processor, add ground spices, garlic, sun-dried tomatoes, mint leaves, cilantro, cayenne pepper, lemon juice, and salt. Process until a thick paste forms, scraping down the sides as necessary.
With the motor running, drizzle in the olive oil until combined. Transfer to an air-tight container and refrigerate for up to 4 days or freeze up to 2 months (see notes).