In a large, heavy pot, heat milk to a gentle boil over medium-high heat, stirring often to prevent scorching (this will take about 30 minutes, so bring a book). As soon as it boils, remove from heat and drizzle in lemon juice, stirring slowly and gently. Keep stirring until solid white curds separate from greenish white, translucent liquid whey. (If whey is still milky instead of clear, return to heat until whey is clear.) Let sit until curds have settled below whey, about 15 minutes.
Meanwhile, line a large colander with cheesecloth and set in sink. Pour curds into colander and rinse gently with lukewarm water 5 seconds. Gather cheesecloth up over curds and gently twist to squeeze out excess liquid (it will still be dripping).
Put a plate on cheesecloth-wrapped curds and top with a 5-lb. weight. Let drain 45 minutes. (At this point it may still be dripping a bit; this is okay.)
Unwrap cheese and put in bowl of stand mixer with dough hook attachment with salt and chives. (You can also knead it by hand.) Beat cheese on medium-low speed or knead it until silky looking and no longer grainy (similar to cream cheese), 10 to 12 minutes.
Roll cheese into a 2-in.-thick log and wrap in waxed paper and then plastic wrap. Refrigerate until cold and firm, at least 1 hour.
Note: Nutritional analysis is per 1/4-in. slice.