In a stand mixer fitted with the dough hook, or in a bowl with a wooden spoon, combine the flour, sugar, yeast and salt. Add the water and eggs. Mix just until the dough starts to form.
Add ¾ cup (170 g) of the butter. Knead the dough for 5 minutes with the dough hook or with your hands on a lightly floured work surface. The dough will be soft and slightly sticky. Place in a clean, lightly oiled bowl. Cover with a clean damp cloth or plastic wrap. Let rise in a warm, humid spot for 1 hour 30 minutes or until doubled in volume.
In a bowl, combine the brown sugar and the cinnamon. Butter a 13 x 9-inch (33 x 23 cm) baking dish.
On a generously floured work surface, roll out the dough to form a 20 x 12-inch (50 x 30 cm) rectangle. Brush with the remaining butter and cover with the brown sugar mixture. Starting on one of the short sides, loosely roll up the dough to get a 12-inch (30 cm) long cylinder, like a cake roll. Cut into 8 slices.
Arrange the slices cut side up in the baking dish without letting them touch. Cover with plastic wrap and let rise in a warm, humid spot for 1 hour or until doubled in volume. At this stage, you can refrigerate the dough for up to 12 hours. In this case, let the dough sit at room temperature for 30 minutes before baking.
Meanwhile, using a pencil, trace a 12 x 6-inch (30 x 15 cm) rectangle on a piece of parchment paper. Turn the paper over and place on a baking sheet.
In a small pot, bring the cream, brown sugar and corn syrup to a boil. Simmer until the mixture is an amber colour, about 4 minutes. Remove from the heat. Stir in the butter with a wooden spoon.
Pour the caramel inside the traced rectangle on the parchment paper. Place another piece of parchment paper over it. Using a rolling pin, roll the caramel between the two sheets of paper to get a rectangle roughly the size of the one you traced. Remove the top layer of parchment paper. Cut the caramel into 8 squares.
With the rack in the middle position, preheat the oven to 350°F (180°C).
Place a square of caramel over each sticky bun. Bake for 25 minutes or until golden. Let cool on a wire rack.
In a bowl, cream the cream cheese and butter with an electric mixer. On low speed, gradually add the icing sugar until smooth and creamy. Spread the icing over the sticky buns, to taste. The sticky buns are best enjoyed the same day they are made.