Place dough on a lightly floured work surface; cover and let stand until butter is slightly softened, 5 to 10 minutes.
Roll dough out to a 10-by-18-inch rectangle. Cut rectangle in half crosswise. Wrap one half with plastic wrap and transfer to refrigerator. Place remaining half on a large (about 18-by-25-inch) piece of parchment paper. Roll into a 16-by-20-inch rectangle, stopping to chill dough as necessary if butter becomes too soft; transfer to a large baking sheet and cover with plastic wrap. Transfer baking sheet to refrigerator and chill until firm, about 30 minutes. Repeat process with remaining piece of dough.
Transfer one piece of dough to a lightly floured work surface so that one of the longer sides is facing you. Halve dough to make two 16-by-10-inch rectangles.
Using a pizza cutter, trim a scant 1/8 inch from all sides of each rectangle of dough. Working quickly, using pizza cutter to cut each half into 1 1/2-by-3-inch rectangles. You should have about 60 rectangles.
Place 1 slice ham in the center of each dough rectangle; top each with a slice of cheese. Starting from one short end and working toward the other, roll to enclose. Brush edges with water and gently press to adhere.
In a small bowl, whisk egg together with 1 teaspoon water. Line baking sheets with parchment paper; place rectangles on prepared baking sheets, seam-side down, leaving room for croissants to rise. Brush tops with egg mixture, reserving remaining egg mixture. Place 3 narrow drinking glasses upside-down between each roll of dough; cover well with plastic wrap, making sure plastic does not touch dough. Let stand in a warm place until doubled in size and feel hollow when gently touched, 2 1/2 to 3 hours.
Preheat oven to 425 degrees.
Uncover and remove glasses from baking sheets; brush croissants lightly with remaining egg mixture (reserving remaining for second piece of dough). Transfer to oven and bake until puffed and golden, about 15 minutes. Transfer to a wire rack to cool. Repeat steps 3 through 8 with remaining piece of refrigerated dough.