In a large skillet, heat 1 1/2 teaspoons butter and 1 tablespoon oil over medium. Add onion and cook, stirring occasionally, until golden, 10 minutes. Add carrot, thyme, and cayenne and season with salt and pepper. Cook, stirring occasionally, until carrot is tender, 5 minutes. (If mixture is dry or begins to overbrown, add up to 2 tablespoons water.) Transfer mixture to a food processor.
Return skillet to medium-high and add remaining 1 1/2 teaspoons butter and remaining 1 tablespoon oil. Pat livers dry with paper towels. Season livers with salt and pepper and add to skillet. Cook until golden brown on all sides and cooked through, 6 minutes. Transfer livers to processor and pulse, scraping down sides as needed, until mixture is mostly smooth. Season to taste with salt and pepper and let cool. Transfer pate to a small bowl and serve at room temperature with toasts or crackers.
To store, press plastic wrap directly on surface and refrigerate, up to 2 days.