Preheat oven to 450°F. Line a rimmed baking sheet with parchment paper or a silicone mat.
Roast red pepper halves cut-side down for 30–35 minutes, until skins blister and blacken.
Transfer hot peppers to a heat-safe bowl and cover to steam for 10–15 minutes.
Peel off the blackened skins under cool running water.
Add peeled peppers, feta, red pepper flakes, and olive oil to a food processor. Pulse until mostly smooth.
Taste and season with salt and pepper. Pulse briefly to combine.
Transfer to a serving bowl, garnish, and serve immediately with dippers of choice.