In a large bowl, whisk together the flour, cloves, cinnamon, ginger, and baking soda; set aside. In another large bowl, beat together the butter, brown sugar, and golden syrup using a handheld mixer set to medium speed until the mixture is pale and fluffy, 1–2 minutes. Add the reserved spice mixture and the heavy cream in 3 alternating batches, beginning and ending with the spice mixture, until the dough just combines. Transfer dough to a work surface, divide in half, and shape each half into a flat disk. Wrap each disk in plastic wrap; refrigerate for 1 hour.
Heat oven to 350°. Unwrap 1 disk of dough and place on a floured work surface. Using a rolling pin, roll dough to a 1⁄8" thickness. Cut out cookies using the cookie cutters of your choice and place cookies 2" apart on parchment paper–lined baking sheets. Repeat with remaining dough, rerolling scraps. Refrigerate for at least 20 minutes. Bake cookies, 1 sheet at a time, until browned and set, about 12 minutes. Transfer cookies to a wire rack and let cool.
To make an icing, if you like, whisk confectioners' sugar, lemon juice, and egg white in a medium bowl until smooth. Transfer icing to a resealable plastic bag (or a pastry bag). Snip off a bottom corner of the bag and pipe icing onto cookies in a decorative pattern.