Pour warm (not hot!) water into the bowl of a stand mixer fitted with the dough hook attachment. Add yeast, sugar, salt, oil, and flour.
Knead on low-medium speed for about 4 minutes, until a smooth dough forms. The dough should be thick, yet soft and only slightly sticky. When it pulls away from the sides of the bowl it's ready.
Place the dough in a greased bowl and turn the dough over to coat all sides. Cover the bowl with plastic wrap and place it in a warm environment to rise until doubled in size, about 1 hour.
Once doubled in size, punch the dough down to release any air bubbles. Remove dough from the bowl and turn it out onto a lightly floured surface. Divide the dough into 6 equal pieces.
Heat oven to 350°F (180°C). Line your cookie sheet with parchment paper, a silicone baking mat or lightly grease the cookie sheet.
Roll each ball of dough into a rope about 14 inches long. Cut a 1/2-inch piece off the end, and roll into a ball for the tail. Fold the dough rope in half, and loosely twist twice to form the body of the bunny. Place on cookie sheet and press down the overlapping ear a little bit to prevent it from rising too high. To make the bunny tail press ball of dough into folded end of the twist. Repeat with remaining dough to make 6 bunnies. Let the bunnies rise for 5-10 minutes.
Brush the tails with egg wash and sprinkle with coarse sea salt.
Bake the bunnies for 23-25 minutes until the tops are golden brown and the edges look cooked. Remove the rolls from the oven when they are done and brush butter onto each warm bunny.