DRY THE FILLETS WITH PAPER TOWELS. (USE THE BACK OF A KNIFE TO SQUEEGEE OFF THE SKIN SIDE.)
PEEL AND TRIM SHALLOT. MINCE AND RESERVE.
HEAT FRYING PAN OVER MEDIUM HEAT. ADD A SPLASH OF OLIVE OIL.
WHEN THE OIL STARTS TO SMOKE, SALT AND PEPPER THE FISH. CAREFULLY LAY THE FILLETS IN THE PAN SKIN SIDE DOWN. SHAKE THE PAN SO THE FILLETS DON'T STICK. CONTINUE SHAKING GENTLY FOR ANOTHER MINUTE. THEY SHOULD SLIDE AROUND THE PAN.
REDUCE THE HEAT TO MEDIUM-LOW AND COOK FOR A MINUTE OR TWO, DEPENDING ON THE THICKNESS OF THE FILLETS.
TOSS IN THE THYME SPRIGS. THEY'LL CRACKLE A BIT WHICH IS GOOD. ADD 3 TBSPS OF BUTTER AND AS SOON AS IT MELTS START BASTING THE FISH. BASTE FOR AROUND 30 SECONDS.
COVER THE FISH WITH A LID. COOK COVERED FOR 30 SECONDS. UNCOVER AND BASTE. CONTINUE THIS WAY TILL THE FISH IS DONE: TEST BY INSERTING A PARING KNIFE INTO THE THICKEST PART OF THE FILLET FOR A FEW SECONDS, THEN TOUCHING IT TO YOUR BOTTOM LIP. IF THE BLADE IS WARM, THE FISH IS DONE.
PLATE THE FISH.
SAUCE: TOSS IN THE CHOPPED SHALLOTS. COOK FOR 30 SECONDS, STIRRING WITH A WOODEN SPOON. POUR IN 1/2 CUP OF LIQUID (WHITE WINE, WATER, CHICKEN OR FISH STOCK, SQUEEZE OF LEMON. CONTINUE STIRRING AND SCRAPING UP THE BROWN BITS TILL THE SAUCE HAS REDUCED AND THICKENED.
TURN OFF THE HEAT AND STIR IN 3 TSPS OF BUTTER. SPOON OVER THE FILLETS AND SERVE.