Combine the sourdough starter, water, flour, melted butter, honey, and salt in a mixer fit with a knead attachment. Combine and knead together until the dough is smooth and sticks to the attachment. Remove from the attachment and allow it to sit in the bowl and rise overnight (or until it doubles in size). Cover the bowl with a damp towel.
After the dough doubles, add the eggs. The eggs may be difficult to fully incorporate but it doesn’t have to be perfect. Use the knead attachment and mix with your hands occasionally to get the egg incorporated. Once it's mostly incorporated, start shaping your buns. The dough will be wet. Section off equal pieces of dough and fold over the edges on itself to shape the buns.
Preheat oven to 350F and line two baking trays with parchment paper. You will have 8 to 10 equal sized buns. Bake for 20 to 30 minutes until the tops are golden brown.