Preheat oven to 400°. Place 1 head of garlic, ¼" cut off top to expose cloves, cut side up, in a corner of a parchment-lined rimmed baking sheet; drizzle with extra-virgin olive oil.

Place 1½ lb. mixed onions (such as sweet, yellow, and/or red), halved, thinly sliced, on baking sheet. Drizzle ¼ cup extra-virgin olive oil and 2 Tbsp. water over and toss until evenly coated. Season with kosher salt and toss again to combine. Mound onions in center of baking sheet (not spreading them out will allow them to steam and soften without browning too quickly). Roast, tossing and stirring onions every 20 minutes, until golden brown and softened, 40–50 minutes (some onions will brown more than others, which is okay). Let cool on baking sheet at least 15 minutes.

Transfer onions and garlic to a cutting board and squeeze cloves out of their skins. Using a flat side of a chef’s knife, smash garlic to a paste and transfer to a medium bowl. Finely chop onions and transfer to bowl with garlic. Finely grate zest from 1 lemon over onions and garlic.

Cut lemon in half and squeeze juice from 1 half into a blender; cover and chill remaining lemon half for another use. Add 1½ cups whole-milk cottage cheese (about 12 oz.), 2 Tbsp. crème fraîche, remaining 2 Tbsp. extra-virgin olive oil, and 2 Tbsp. water to blender. Cover and blend, stopping motor and scraping down sides of jar if needed, until smooth. Using a rubber spatula, scrape cottage cheese mixture into bowl with reserved onion mixture. Add ½ tsp. Diamond Crystal or ¼ tsp. Morton kosher salt and ¼ tsp. freshly ground pepper and mix well.

Transfer dip to a small bowl and sprinkle with cracked pepper. Serve with potato chips for dipping.