In a large mixing bowl, whisk together the flour, instant yeast, sugar, and salt.
Pour in the warm water and whisk hard for 2 minutes until you have a smooth, thick batter with no lumps. It should be thicker than pancake batter but still pourable off a spoon.
Cover the bowl with a damp towel and set it in a warm spot for 60 minutes, until the batter has doubled and looks full of bubbles.
While the batter rises, make the syrup: combine the honey, water, lemon juice, and cinnamon stick in a small saucepan. Bring to a simmer over medium heat and cook for 3 to 4 minutes, until slightly thickened. Set aside off heat.
Pour the vegetable oil into a 26 cm / 10-inch heavy pot to a depth of about 7 cm / 3 inches. Heat over medium heat to 180C/350F, checking with a thermometer.
Dip a tablespoon in a bowl of water, scoop a rounded tablespoon of batter, and slide it into the hot oil. Repeat with 8 to 10 more spoonfuls, wetting the spoon between scoops so the batter releases cleanly. Do not crowd the pot, or the doughnuts will stick together and cook unevenly.
Fry for 3 to 4 minutes, turning once or twice with the slotted spoon, until deep golden brown and puffed. The doughnuts should sound hollow when tapped and float high in the oil.
Lift the doughnuts out with the slotted spoon and drain on a paper-towel-lined tray. Let the oil come back to 180C/350F between batches and repeat with the remaining batter.
While still warm, dip each doughnut in the honey syrup for a few seconds, or arrange them on a platter and spoon the warm syrup over the top.
Sprinkle immediately with the crushed walnuts, ground cinnamon, and sesame seeds if using. Serve warm.