In a small bowl, microwave the milk and sugar for 45 seconds, until it is warm but not hot. Add in yeast and stir. Allow to sit for 5 minutes. The mixture should foam, signaling that the yeast is active. Then, stir in melted butter, oil, eggs, vanilla extract, and salt.
Pour mixture into a standing mixer fitted with a dough hook. Turn the mixer to low speed and slowly add the flour 1 cup at a time. Toss in poppyseeds. Let the dough knead for 5 minutes. Dough should be slightly sticky, but firm enough to hold together in a ball.
Coat a large bowl with oil and place the dough into the bowl. Cover with plastic wrap and set in a warm area to proof for 1 hour or until the dough has doubled in size.
Line an 8″x8″ pan with parchment paper. Set aside.
Flour a large work surface. Roll the dough out into a 10 inch by 14 inch rectangle. Smear the butter over the dough, coating it evenly. Sprinkle the sugar and lemon zest on top. Make sure to spread everything evenly to the edges.
Roll the dough into a log, pinching the ends so it stays together. Using a serrated knife, cut the dough into 3/4 inch rolls/buns. Place 9 buns into the pan. You can still bake your leftover buns (you should have 2-3) by putting them into a large ramekin or muffin tin. They’re extras!
Allow dough to rise a second time. Cover the pan with a cloth and let rest in a warm place for 45 minutes.
Preheat the oven to 375°F. Bake for approximately 30 minutes, until the top is nicely golden brown. You’ll want a deep golden color, as the outside of the loaf can be cooked and the inside still doughy. Use your discretion.
While the buns are baking, mix all frosting ingredients together. The consistency should be thick. Once out of the oven, allow the buns to cool for 10 minutes, then top with frosting.