Place the dried chiles in a large heatproof measuring glass. Pour boiling water over to submerge, use a small plate to hold down the chilis, if needed. Let sit until chiles are very pliable and cool enough to handle, 15–20 minutes. Drain; remove stems and seeds and discard.
Toast cumin and coriander in a dry small skillet over medium-low heat, tossing constantly, until very fragrant, about 3 minutes.
Transfer to a food processor, add garlic, and pulse until spices are broken up and garlic forms a paste. Add chiles and pulse until chiles form a coarse paste. Add lemon juice, vinegar, paprika, and salt and process until mostly smooth but mixture still has a little texture.
With the food processor running add olive oil slowly. Process until the oil is incorporated. Store the paste in a jar, top with a thin layer of olive oil.
Store the harissa paste in the refrigerator. Top with a thin layer of olive oil after each use to extend the shelf life. When stored in this manner the harissa should last up to 2 months.