HEAT 1 CUP MILK IN A SMALL SAUCEPAN OVER MEDIUM HEAT UNTIL IT REGISTERS 110 DEGREES ON A CANDY THERMOMETER. POUR MILK INTO A MIXER BOWL, AND FIT MIXER WITH A DOUGH HOOK. WITH MIXER ON LOW SPEED, ADD GRANULATED SUGAR, YEAST, BUTTER, 1 1/2 TEASPOONS SALT, THE NUTMEG, CINNAMON, AND EGGS. ADD FLOUR, 1 CUP AT A TIME, AND KNEAD UNTIL MIXTURE COMES TOGETHER IN A SOFT, STICKY DOUGH. CONTINUE KNEADING, SCRAPING DOWN HOOK AS NEEDED, UNTIL DOUGH IS SMOOTH, ABOUT 4 MINUTES.

ADD CHERRIES AND RAISINS, AND KNEAD TO INCORPORATE. TURN DOUGH ONTO A LIGHTLY FLOURED SURFACE, AND KNEAD TO DISTRIBUTE DRIED FRUIT. COAT A LARGE BOWL WITH BUTTER. SHAPE DOUGH INTO A BALL, AND PLACE IN PREPARED BOWL. COVER WITH A PIECE OF PLASTIC, AND LET RISE UNTIL DOUBLED IN SIZE, ABOUT 1 HOUR.

GENEROUSLY BUTTER A RIMMED BAKING SHEET. TURN DOUGH ONTO A SURFACE, KNEAD BRIEFLY, THEN DIVIDE INTO 3 PIECES. WORKING WITH 1 PIECE AT A TIME, DIVIDE EACH THIRD INTO 10 PIECES, AND SHAPE EACH INTO A TIGHT BALL. (KEEP DOUGH COVERED WITH PLASTIC.) PLACE ON PREPARED SHEET, SPACING 1/2 INCH APART. REPEAT WITH REMAINING DOUGH. COVER WITH PLASTIC, AND LET RISE IN A WARM SPOT UNTIL BUNS HAVE DOUBLED IN SIZE AND ARE TOUCHING, ABOUT 1 HOUR.

PREHEAT OVEN TO 375 DEGREES. WHISK TOGETHER EGG WHITE AND WATER IN A SMALL BOWL. BRUSH TOPS OF BUNS WITH EGG-WHITE WASH. BAKE, ROTATING SHEET HALFWAY THROUGH, UNTIL GOLDEN BROWN, 20 TO 22 MINUTES. LET COOL ON SHEET ON WIRE RACK FOR 30 MINUTES.

WHISK TOGETHER REMAINING 3 TABLESPOONS PLUS 2 TEASPOONS MILK, CONFECTIONERS’ SUGAR, VANILLA, AND A PINCH OF SALT. SPOON ICING INTO A PASTRY BAG FITTED WITH A PLAIN 1/4-INCH ROUND TIP, AND PIPE ICING ON BUNS IN THE SHAPE OF A CROSS. (ALTERNATIVELY, SPOON GLAZE ON BUNS.) SERVE IMMEDIATELY.