Preheat oven to 350 degrees F (175 degrees C). Line a baking sheet with parchment paper.
Warm 1 cup milk in a large saucepan over low heat until it reaches room temperature, 3 to 5 minutes. Remove from heat and mix in yeast until dissolved. Mix in egg, olive oil, sugar, and salt.
Stir half of the flour into the saucepan. Add carrots and sunflower seeds. Stir in remaining flour until dough is sticky. Cover and let rise in a warm place for 30 minutes.
Divide dough into 12 pieces; roll into small buns. Arrange buns on the prepared baking sheet. Cover and let rise until puffy, about 15 minutes.
Brush tops of buns with 1 tablespoon milk.
Bake in the preheated oven until golden brown, 15 to 17 minutes.