Dissolve the yeast in the lukewarm water and let sit about 5-10 minutes until foamy.
Using a mixer fitted with a whisk attachment, combine the yeast mixture, oil, egg, and sugar in a large bowl. Add the remaining 3/4 cup of water and mix to combine.
Switch to the the dough hook attachment and add the flour. Mix on medium speed until the dough holds together to become a cohesive, slightly sticky ball (about 5 minutes). When you lift up the mixer-head the dough should stick together as one mass of dough, however, it should stretch from the dough hook to the bottom of the bowl. (See photo above). Let the dough rest for 20 minutes in the bowl.
Sprinkle the salt over the top, and mix on medium speed for another minute until fully incorporated.
Transfer the dough to a lightly-oiled, large bowl. The dough will be fairly sticky and some will stick to the original bowl, and to your fingers when transferring.
Cover it with a clean tea towel or lid and let it rise in a warm place**** until doubled in size, about 1-1.5 hours. Once risen or proved, punch down the dough once to remove the air.
Line a 9x13 pan with parchment paper (or forego the parchment paper by lightly greasing the pan). Coating your fingers in some flour, begin to pinch off and roll the dough into golf-ball-sized balls of dough. Place dough balls slightly touching in the pan. This recipe will make 15 rolls (3 rows of 5 rolls).
Cover with a tea towel and allow to rest in the same warm place for about 20 minutes, until they become lighter and have the consistency of soft marshmallows. They will not fully double in size on the second rise.
Bake the rolls at 350℉ for about 25 minutes. Remove from oven when the tops are slightly golden. Brush with softened or melted butter and serve immediately.