Preheat oven to 425°. On a parchment-lined rimmed baking sheet, toss carrots with 2 tbsp. oil, the chile, and a generous pinch or two of salt; spread in an even layer. Roast, tossing once, until carrots are tender and beginning to caramelize, 20 to 25 minutes. Let carrots cool slightly on baking sheet.
Discard chile. Transfer carrots with their oil to a food processor. Add a pinch of salt, the garlic, yogurt, vinegar, and remaining 2 tbsp. oil. Purée until very smooth, scraping down sides of bowl as needed, about 4 minutes. Taste and add more salt or acid (yogurt, vinegar, or lemon juice) as needed to balance the carrots’ sweetness.
MAKE AHEAD: Up to 3 days, chilled airtight.
Makes about 2 cups