PEEL THE KOHLRABI AND POTATO, CUT INTO CHUNKS AND PLACE IN A MEDIUM SAUCEPAN. POUR IN MILK TO BARELY COVER.
BRING TO SIMMER ON A LOW FLAME. ADD SALT AND A FEW GRINDINGS OF WHITE OR BLACK PEPPER.
SIMMER FOR 10-15 MINUTES, WATCHING CAREFULLY BECAUSE THE MILK WILL BOIL OVER FASTER THAN YOU CAN BLINK AN EYE. THE VEGETABLES ARE DONE WHEN EASILY PIERCED WITH A SHARP KNIF.
SPOON POTATO AND KOHLRABI INTO A FOOD PROCESSOR AND PURÉE, ADDING AS MUCH MILK FROM THE PAN AS YOU NEED.
HEAT A FRYING OVER HIGH HEAT FOR A MINUTE. ADD A GENEROUS SPLASH OF OLIVE OIL.
TRIM AND PEEL THE ONION. SLICE LENGTHWISE IN HALF, THEN SLICE EACH HALF CROSSWISE INTO THIN HALF MOONS.
CUT THE TOMATOES LENGTHWISE IN HALF AND PLACE IN A BOWL. STRIP THE LEAVES FROM THE THYME SPRIGS AND ADD TO THE TOMATOES.
DRY THE FILLETS SCRUPULOUSLY WITH PAPER TOWELS. SEASON WITH SALT AND PEPPER. WHEN THE OIL IN THE PAN IS SMOKING, CAREFULLY LAY IN THE FILLETS, SKIN SIDE DOWN. GENTLY SHAKE THE PAN SO THE FILLETS SLIDE AROUND AS THEY BROWN.
BASTE THE FISH WITH THE OLIVE OIL. COVER THE PAN REMOVING ONLY TO BASTE THE FILLETS. DEPENDING ON THEIR THICKNESS, COOK THE FISH FOR 4-6 MINUTES. TEST FOR DONENESS BY INSERTING A SHARP KNIFE IN THE CENTER OF THE FILLET AND TOUCHING IT TO YOUR BOTTOM LIP. IF IT'S WARM, THE FISH IS DONE. PLACE FILLETS ON FOUR WARMED PLATES.
ADD A SPLASH MORE OF OLIVE OIL. RAISE THE HEAT AND ADD THE ONIONS. SAUTÉ TILL THEY START TO SOFTEN. ADD THE TOMATOES AND THYME AND A PINCH OF SAFFRON. SEASON WITH SALT.
COOK, STIRRING WITH A WOODEN SPOON, TILL THE TOMATOES BEGIN TO BREAK DOWN. ADD 1/2 CUP OF WATER OR WHITE WINE. STIR WITH A WOODEN SPOON OVER HIGH HEAT, SCRAPING ANY BITS WHICH STICK TO THE SIDES OR BOTTOM OF THE PAN. ONCE THE LIQUID HAS REDUCED TO ABOUT A 1/4 CUP, TURN OFF THE HEAT AND QUICKLY STIR IN 2 TBPS OF BUTTER IN BITS. THIS WILL GIVE YOUR SAUCE A VELVETY SHEEN.
SPOON OVER THE FISH. SPOON PURÉE ONTO EACH PLATE. GARNISH WITH FRESH PARSLEY LEAVES AND SERVE.