Mix 1 cup lukewarm water, sugar, and yeast together in a large bowl. Let stand until yeast softens and forms a creamy foam, about 20 minutes.
Stir soy milk, 1/2 cup soy butter, avocado oil, 6 tablespoons water, egg replacer, and salt gently into the yeast mixture using a fork. Mix in 2 cups bread flour. Continue mixing by hand, adding flour 1/2 cup at a time, until dough comes together. Knead dough in the bowl for 2 minutes. Cover with plastic wrap and a warm towel; let rise for 1 hour 30 minutes.
Divide dough into 24 pieces; knead and form into balls.
Grease 2 muffin tins with cooking spray. Place 1 ball of dough in each muffin cup. Cover with plastic wrap and warm towels; let rise for 1 hour.
Preheat oven to 425 degrees F (220 degrees C).
Melt remaining 1/4 cup soy butter in a small saucepan over low heat, about 2 minutes. Remove from heat; whisk in honey.
Bake rolls in the preheated oven until puffed and pale, about 8 minutes. Remove from the oven; brush with half of the soy butter and honey mixture. Return to the oven and bake until golden brown, about 9 minutes more. Brush with remaining soy butter and honey mixture.