Stir together 1 cup lukewarm water and sugar in bowl of a stand mixer until sugar has dissolved. Sprinkle with yeast, and let stand until bubbling and foamy, about 5 minutes. Stir in 2/3 cup sweetened condensed milk and salt. Add flour, and stir to form a shaggy mass. Let stand to hydrate, uncovered, 5 minutes.

Attach dough hook to stand mixer. Beat mixture on medium speed until dough is relatively smooth and elastic but still sticky, about 3 minutes. Using a flexible bench scraper, scrape dough onto a work surface generously dusted with flour. Using scraper, flip and lightly dust dough with flour to make it easier to handle. Swiftly and gently knead dough using your hands to form a smooth, round shape, about 30 seconds. Press dough with a finger; the slightly sticky, soft dough will gently spring back, leaving a shallow indentation. Drizzle vegetable oil around sides and bottom of a medium bowl. Transfer dough to greased bowl, rotating dough to fully coat with oil. Cover bowl with plastic wrap, and let dough rise in a warm (80°F to 90°F), draft-free place until it has nearly doubled in size, about 1 hour.

Punch down dough in bowl. In a clockwise direction, pull up the outer edge of the dough ball and fold into the center, 4 to 6 folds total. Flip dough over so folded side faces bottom of bowl. Cover bowl with plastic wrap, and let dough rise in a warm place until nearly doubled in size, 30 to 45 minutes.

Line a 13- x 9-inch rimmed baking sheet with parchment paper, leaving a 1-inch overhang on long sides. Remove plastic wrap from dough; set plastic wrap aside. Turn dough out onto a lightly floured work surface. Divide dough into 2 equal portions. Working with 1 dough portion at a time, shape into a rope about 10 inches long and 2 inches wide. Cut each rope crosswise into 10 even pieces. Swiftly separate pieces; roll each piece into a roughly 2-inch ball. Pull, tuck, and gather edges of each ball toward center to form a small, taut sphere; if sphere is uneven, cup your hand over dough, and rotate it 3 or 4 times on work surface to round out. Arrange balls, seam sides down, side by side and barely touching, on prepared baking sheet, making rows of 5 doughballs. Repeat process with remaining dough half, adding rolls to same baking sheet to form 20 rolls total. Using your palms, press down firmly on dough balls until they touch and flatten to about 3/4 inch tall. Loosely cover using reserved plastic wrap. Let rise in a warm place until nearly doubled in size, 45 minutes to 1 hour. During final 30 minutes of dough rise time, preheat oven to 350°F.

Stir together remaining 1 1/2 tablespoons sweetened condensed milk and 1 tablespoon tap water in a small bowl. Brush tops of risen rolls with about half of diluted condensed milk; set aside remaining diluted condensed milk. Bake in preheated oven until tops of rolls are richly browned, 16 to 20 minutes, rotating baking sheet on oven rack 180 degrees when rolls are light golden, after about 12 minutes. Remove from oven, and let rolls cool on baking sheet on a wire rack 3 to 5 minutes. Brush rolls with reserved diluted condensed milk to glaze. Let cool 20 minutes. Using parchment paper overhang as handles, lift parchment paper from baking sheet to remove rolls from baking sheet. Transfer parchment paper with rolls to a wire rack. Enjoy rolls warm or at room temperature.