To make ricotta: Combine milk, buttermilk, and salt in a large, nonreactive pot and bring to a gentle simmer. Cook, without boiling, for 5–8 minutes. Meanwhile, line a strainer with a double thickness of cheesecloth. Pour curds (solids) and whey (liquids) through strainer. Let mixture drain for 1 hour. Transfer mixture to a bowl and fold in lemon zest, garlic, pepper, and herbs. Keep ricotta for up to 5 days in the fridge, in a covered container. (Makes about 2 1/4 cups.)
In a medium bowl, combine Dijon and lemon juice; season with salt and pepper. Whisk in olive oil. Set aside.
Preheat grill. Create one 10-inch pizza dough crust. Grill dough on one side for about a minute. Flip dough, pop any air bubbles, and spread with ¾–1 cup ricotta. Cover grill. After 3–4 minutes, check crust bottom. If it’s a little charred and dough looks set, it’s ready.
When ready to serve, toss arugula with about one-fourth of the dressing or to taste, plus flaxseed and chives. Sprinkle over grilled pizza and garnish with grated Parmesan and red onion slivers, if desired.