In small saucepan, bring balsamic vinegar and sugar to boil, stirring. Reduce heat and simmer until syrupy, about 10 minutes. Transfer to small plastic squeeze bottle or jar and let cool. (Make-ahead: Store in cool place for up to 2 weeks.)
Meanwhile, heat oil in saucepan over medium-high heat to 375°F. Set aside half of the basil leaves; cook remaining basil in 2 batches until crisp, 5 to 6 seconds. Transfer to paper towel–lined plate; set aside. Let oil cool.
Reserving 2 tbsp of the oil to cook pickerel, strain remaining oil into food processor; add remaining basil and pulse until finely chopped. Strain oil through fine sieve and funnel into separate small plastic squeeze bottle. (Make-ahead: Refrigerate for up to 1 week.)
In pie plate, whisk together flour, lemon rind, salt and pepper. With paper towel, pat fillets dry; dredge in flour mixture to coat well on both sides, shaking off excess. Set aside in single layer on waxed paper–lined baking sheet.
Place large nonstick skillet over medium heat; brush generously with reserved basil oil. Fry pickerel, in batches if necessary and brushing with more oil as necessary, until golden and flakes easily when tested with fork, 3 to 5 minutes per side.
Divide fish among 4 warmed plates; top each with pile of fried basil leaves. Drizzle basil oil around fillet. Dot oil with balsamic syrup.