Add dried chilies to a mixing bowl or measuring cup and cover with hot water. (We used a mix of mild New Mexico chiles and hot peppers (chiles de arbol). Adjust the ratio to preferred spice level.) Once submerged, cover and steam for 15-20 minutes to rehydrate. Set aside.
In the meantime, add cumin and coriander seeds to a small skillet and toast over medium heat for a few minutes, or until fragrant and they start to pop. Then transfer to a mortar and pestle (if you don't own one, just add to a food processor to mix later), and crush into a fine powder.
Next add caraway seeds, minced garlic, smoked paprika, and salt, and mix. Then add lemon juice and vinegar and mix again until you've achieved a paste. Transfer paste to food processor and add the tomato paste. Set aside.
Once chiles are rehydrated, drain and remove the stems and seeds (I recommend wearing gloves to protect your hands), then add to food processor. At this point, all ingredients (besides olive oil) should be in the food processor.
Blend for 1-2 minutes, scraping down sides as needed until a smooth paste is achieved. Then stream in 1/4 cup olive oil while blending to create a saucier consistency (see photo). (Add more oil (or water) as needed until desired consistency is achieved.)
Taste and adjust flavor as needed, adding more lemon or vinegar for acidity, paprika for smokiness, tomato paste for depth of flavor, or salt to taste. If it's not spicy enough at this point, you can either add more chiles de arbol (rehydrated and seeds removed) or cayenne pepper to taste.
Scoop paste into a jar to store in the refrigerator for up to 1 month. If storing in the freezer, we recommend scooping into an ice cube tray, freezing, and storing in a freezer-safe container up to 2 months. Add cubes directly to warm foods (like stews or cooked sauces). If adding to cold foods (like dressings or dips), let thaw before adding.
*Adapted from Bon Appetit. Find a more traditional version here! *Nutrition information is a rough estimate.