Preheat the oven to 350°F. Grease a 9-by-13-inch baking pan with butter.

Make the cake: Pick out 24 of the best-looking walnut halves and set them aside for garnish. Put the remaining walnut halves in the bowl of a food processor, add the sugar, and pulse until the nuts are ground to a coarse meal. Be careful not to overprocess the nuts to a paste.

In a large heavy saucepan (large enough to hold all the batter ingredients), combine the water, butter, honey, cinnamon, salt, and ground walnut mixture and cook over medium-high heat, stirring frequently, until the butter melts and the mixture is uniformly combined. Remove from the heat.

Put the flour, baking powder, and baking soda in a bowl and whisk until blended, then stir into the honey-walnut mixture until just blended.

Pour the batter into the prepared pan and arrange the reserved walnut halves in even rows on top of the cake. It's fine if they sink somewhat into the batter, nut they should still be fully visible.

Bake the cake until the top is golden brown and firm to the touch and a skewer inserted in the center comes out clean, or an instant-read thermometer inserted in the center reads at least 185°F, 30-40 minutes. Place the pan on a wire rack and let the cake cool to lukewarm.

Meanwhile, make the syrup. Combine the honey, water, and lemon zest in a small saucepan, bring to a simmer over medium-high heat, and cook until the mixture becomes syrupy, about 5 minutes. Remove from the heat and stir in the lemon juice.

While the cake is still warm, spoon the syrup evenly over the top, letting each addition sink in before you pour on the next spoonful. Let the cake sit for at least 4 hours, or, better yet, overnight, to absorb the syrup.

Cut the cake into squares (with a walnut half in the center of each). Leftovers can be stored tightly wrapped or in an airtight container for up to 5 days.