Stir together the warm water, yeast, and 1 teaspoon of the sugar in a small bowl. Let the mixture stand until foamy, about 5 minutes.

Whisk together the yeast mixture, milk, olive oil, salt, and remaining ⅓ cup sugar in a large bowl. Add the flour, and stir until fully combined and smooth. Cover the bowl with a clean kitchen towel or plastic wrap, and let it stand in a warm place (about 75°F) until the dough has doubled in size, about 1 hour. (Alternatively, cover the bowl and refrigerate it overnight; let it come to room temperature in a warm place for 30 minutes before frying.)

Pour the oil into a large Dutch oven or pot to a depth of 2 inches. Heat the oil over medium-high until it reaches 330°F. While the oil heats, line a large rimmed baking sheet with several layers of paper towels.

Scoop and fry the loukoumades in batches of 4 to 6. Spray a ½-ounce (2-teaspoon, #60) cookie scoop with cooking spray or dip it in water. Scoop balls of dough, and carefully drop into oil. (Alternatively, pinch off about 1 tablespoon dough, and form into a ball-shape before adding to oil). Fry until golden brown on all sides, 3 to 4 minutes total, flipping a few times during cooking. (Some dough balls may turn on their own.) Remove using a slotted spoon or spider, and place on the prepared baking sheet.

While the loukoumades cook, melt the butter in a small saucepan over medium heat. Add the honey and cinnamon, and stir to combine. Remove from heat, and add a pinch of salt to taste.

Place about half of the loukoumades in a large bowl, and add half of the honey mixture; toss to coat. Transfer to a platter, and repeat the process with remaining loukoumades and honey mixture. Serve immediately sprinkled with walnuts.