To make the pretzels: combine the water and milk in a small saucepan and heat to 112 degrees F. Transfer to the bowl of a standing mixer and add the sugar, honey, and yeast. Stir and let sit until foamy, usually about 5 minutes.

Add the melted butter, egg yolk, salt and 3 3/4 cups of the flour to the mixture and mix on low speed until combined. Increase the speed to medium and continue kneading until the dough is smooth and begins to pull away from the side of the bowl, about 3 to 4 minutes. If the dough appears too wet and either doesn’t pull away from the sides of the bowl or feels too sticky, add additional flour, 1 tablespoon at a time, and continue kneading until it pulls away from the bowl. Remove the dough from the bowl, place on a flat surface and shape the dough into a smooth ball with your hands.

Coat a large bowl with the oil, add the dough, and turn to coat with the oil. Cover with a clean towel or plastic wrap and place in a warm spot until the dough doubles in size, about 1 hour.

Preheat the oven to 425 degrees F. Bring the water to a boil in a large pot.

Remove the dough from the bowl and place on a flat surface. Divide the dough into 6 equal pieces. Roll each piece into a rope about 24 inches long. To shape into pretzels, position the rope like an upright “u” on the work surface. Cross the right side over to the left. Cross right to left again and then bring the dough edges down and push them onto the bottom of the “u” to complete the pretzel shape. Slowly add the baking soda to the boiling water. Boil the pretzels in the water solution, 2 at a time for 30 seconds, splashing the tops with the warmed water using a spoon. Remove with a slotted skimmer or spider. Place the pretzels onto a parchment lined baking sheet, brush the tops with the egg wash and season liberally with sea salt. Bake for 15 to 18 minutes, or until pretzels are deeply golden and cooked through.

Remove pretzels from oven and transfer to a wire cooling rack. Serve warm or at room temperature.