In large bowl, whisk together 1 3/4 cups milk, heavy cream, and eggs. Gradually whisk in flour until smooth. Whisk in remaining 1 cup milk, sugar, salt, and vanilla. Strain batter through fine-mesh sieve, discarding any lumps. Refrigerate at least 1 hour or up to 2 days.
Using pastry brush or paper towel, lightly coat 6-inch crêpe pan or cast-iron skillet with butter. Heat pan over moderately high heat until butter is hot but not smoking. Whisk batter briefly to reincorporate any settled solids.
Ladle about 2 tablespoons batter into pan, immediately tilting and rotating skillet to coat bottom. Cook until crêpe is just set and golden around edges, 10 to 15 seconds. Using tip of knife, loosen edge of crêpe, then, using fingertips, carefully flip crêpe over. Cook until underside is set, about 20 seconds more. Transfer cooked crêpe to platter and keep warm.
Repeat to cook remaining crêpes, coating pan with butter each time and stacking crêpes on platter.
In a sauce pan, bring balsamic to a simmer, turn to low and reduce to half of original amount. Roughly 5-10 minutes. Turn off heat and whisk in cold butter till fully incorporated. Drizzle over crêpes.