Step 1

Heat butter in a small saucepan over medium. Cook onions, stirring often, until onions are very soft but haven’t taken on any color, 14–18 minutes. Add cream and bring to a simmer. Cook, stirring occasionally, until liquid is reduced by half, 6–8 minutes. Let cool slightly, then transfer to a blender and purée until smooth. Season onion purée with salt. Keep warm.

Step 2

Place eggs, breadcrumbs, and flour in 3 separate shallow bowls. Line a rimmed baking sheet with paper towels and set a wire rack inside. Place 1 chicken breast on a cutting board. Holding knife blade parallel to board, cut breast in half, slicing along a long side, to make 2 pieces. Place inside a large resealable plastic bag, seal, and gently pound to about ½" thick. (You aren’t flattening cutlets very much; you just want to make sure they are uniform in shape). Repeat with remaining chicken breast.

Step 3

Working with 1 cutlet at a time, season with salt on both sides, then dredge in flour, shaking off excess. Dip in eggs, letting excess drip back into bowl, then coat in breadcrumbs, pressing to adhere before shaking off excess. Place on a rimmed baking sheet.

Step 4

Heat 2 Tbsp. oil in a large heavy skillet, preferably cast iron, over medium-high. Cook peppers, shaking pan occasionally, until lightly blistered and charred all over, about 5 minutes. Transfer peppers to a medium bowl and toss with lemon juice; season with salt.

Step 5

Wipe out skillet and pour in oil to come 1" up sides. Heat oil over medium-high until an instant-read thermometer registers 350°–365° (don’t let the tip of the thermometer touch the bottom of the skillet). Carefully lower 1 cutlet along the side of skillet closest to you and let slide away from you into the oil. Use tongs to submerge it and cook just until golden brown underneath, about 2 minutes. Carefully turn cutlet and cook, spooning hot oil over any areas that may be poking out of oil, until golden brown on the other side, about 2 minutes. Transfer to wire rack in prepared baking sheet. Season lightly with salt while still hot. Repeat with remaining cutlets.

Step 6

Divide cutlets and warm onion purée among plates and top with charred peppers.

Step 7

Do Ahead: Onion purée can be made 1 day ahead. Let cool; cover and chill.