Stir mayonnaise, lemon juice, garlic, and paprikas together in a bowl. Chill aioli until serving.
Clean fish (steps 1 and 2 of "You Found the Fish," below).
Preheat oven to 200°. Line a rimmed baking sheet with paper towels and keep warm in oven. Fill a large pot with 1 in. oil, insert a deep-fry thermometer, and bring oil to 375° over medium-high heat.
Combine flour and 1/2 tsp. salt in a pie pan, and panko and remaining 1/2 tsp. salt in another pie pan. In a shallow bowl, whisk eggs to blend. Dip fish in flour, shaking off excess, then in egg, then in panko, turning to coat; set on a baking sheet.
Fry one-quarter of fish at a time until golden, 1 to 1 1/2 minutes. Transfer to pan in oven. Serve with aioli and lemon wedges.
You Found the Fish--Now What? Some markets will clean them for you, but if not, you can do it yourself easily enough--it just takes a little practice.
SCALE AND CUT. Scrape off the scales gently with fingertips. Cut through both sides of belly 1/4 in. from edge, from collar to tail.
CLEAN. Pull out the guts; rinse the fish inside and out. Snip off fins.
FILLET (RAW). Score fish all the way around collar just to the bone. Slide your index finger into cut on one side of collar and your middle finger into the other side. Slide your fingers along spine to tail, pulling fillets free. Cut off tail and pull out any remaining bones.
FILLET (COOKED). Make 3 cuts through flesh to bones: along the length of the spine (to one side of it), at collar, and just above tail. Slide knife under fillet to free it from spine, and lift it off. Pull up tail and lift off spine and head to free bottom fillet. Pull out any remaining bones.
*If you can't find fresh anchovies or smelt, you can make the recipe with sardines: Discard heads and tails, and cut bodies into 2-in. pieces. Cut cooked meat from the bones.
Note: Find these little fish at some grocery stores, fish markets (you might need to order them), and Asian markets. If you live near a coastal town, head to the docks-fresh anchovies and smelt are often sold as bait. Look for fish with bright eyes, shiny skin, and a mild aroma. They're very perishable, so plan to cook them the same day.
Note: Nutritional analysis is per serving.