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Combine the milk, butter, and sugar in a microwave-safe measuring cup or bowl, or in a saucepan. Heat to lukewarm, stirring to soften/melt the butter. Pour the mixture into a mixing bowl, and let it cool until it's below 120°F, about 15 minutes.
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Weigh your flour; or measure it by gently spooning it into a cup, then sweeping off any excess. Add the lemon or vanilla, salt, 2 cups of the flour, the eggs, and yeast.
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Beat the mixture on medium speed for about 3 minutes; if you're using a bread machine, program the machine for dough or manual, and mix for several minutes, until everything is well combined.
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Add the remaining flour to the mixture and beat for about 3 minutes, until it becomes a soft, cohesive dough. Note: This is a very sticky, wet dough; it's too sticky to knead, so has to be beaten to develop its gluten.
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Cover the dough and allow it to rise for 45 to 60 minutes, or until almost doubled in bulk. If you're using the dough cycle of a bread machine, simply let the machine complete its cycle.
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Lightly grease the cups of two standard muffin pans (24 cups total). If you don't have two muffin pans (or want to make a loaf, in addition to rolls), lightly grease one muffin pan, plus a 6" soufflé dish or other round, bakeable dish or small loaf pan.
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Divide the dough among 18 cups of the muffin pans, filling each a bit more than half full. If you're using one muffin pan, add dough to each cup then shape the remainder into a loaf, and place it in the small pan of your choice.
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Cover the pans, and let the rolls rise for 45 to 60 minutes, until they're noticeably puffy. Towards the end of the rising time, preheat the oven to 350°F.
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Bake the rolls until they're golden brown and a digital thermometer inserted into the center of one reads at least 190°F, about 15 minutes. If you're making a loaf as well as rolls, bake the loaf for about 25 to 30 minutes, tenting it with aluminum foil after 15 minutes to prevent over-browning.
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Remove the rolls from the oven, and turn them out of the pan onto a rack. Brush with melted butter, if desired. Wrap completely cooled rolls airtight, and store at room temperature for several days; freeze for longer storage.
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Serve rolls with butter and jam, at breakfast or brunch; or as part of an afternoon "high tea."
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Yield: 18 rolls; or 12 rolls and 1 small loaf.