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Rinse rockfish fillets in cold water, then pat dry.
Zest and juice lemon the lemon.
Combine the flour, salt, and pepper on a shallow plate.
Heat 1 of tablespoon butter and the olive oil in a 12-inch nonstick skillet over medium heat.
Dredge the rockfish fillets in the flour mixture one at a time, and add to skillet when butter has melted. Discard the extra flour.
Pan-sear the fillets for 5 minutes, until golden brown on the bottom, then flip to brown the other side. Flatten the fillets gently with a spatula to ensure even browning.
When the second side has browned (about 3-4 minutes) and fillets are cooked through, remove from pan and set aside.
Add the remaining 4 tablespoons of butter to the pan, and turn the heat down to low.
When the butter starts to brown (this should take 1-2 minutes), add 1 tablespoon of lemon juice, 1 tablespoon of lemon zest, capers, and 1 tablespoon of dill, and cook for 30 seconds.
Turn off the heat and taste the sauce, adding more lemon juice, salt, and/or pepper as desired.
Drizzle the sauce over fillets, garnishing with more chopped dill.