Caramelized onions with balsamic glaze:

1. Heat olive oil in a large skillet with a lid over medium-low heat. Add the onions and a pinch of salt, cover, and cook 12–15 minutes, stirring occasionally, until tender.
2. Remove the lid and cook over medium heat, stirring frequently, 3–5 minutes more until golden.
3. Add the balsamic vinegar and cook, stirring, until the liquid evaporates, about 1 minute. Season with the remaining salt and pepper. Remove from heat and let cool. Refrigerate if making ahead.

Pâte à choux (choux pastry):

1. Heat the oven to 425°F (218°C). Line a baking sheet with parchment or a baking mat. Cut the butter into small pieces so it melts quickly and evenly.
2. Combine the butter, water, and salt in a saucepan over high heat. Bring to a rolling boil, making sure the butter is fully melted before the water boils. If needed, reduce heat to melt the butter, then return to a boil.
3. Take the pan off the heat and add all the flour at once. Stir vigorously until a smooth dough forms and no dry flour remains — it should look like mashed potatoes.
4. Return the pan to medium heat. Cook the dough, mashing it against the sides and bottom, then gathering it back into a ball, for 3–5 minutes. The dough is ready when it pulls away from the pan, looks shiny, and holds a spoon upright.
5. Transfer the dough to the bowl of a stand mixer fitted with the paddle attachment. Beat on medium-low until just warm to the touch.
6. Whisk the 4 eggs together. With the mixer running on medium-low, add the eggs in four separate additions, letting each absorb fully before adding the next. After the third addition, check the dough: it should be soft, smooth, and drop from the spatula in a “V” shape. Add the fourth egg (or just half) if needed.
7. Scoop or pipe the dough onto the baking sheet in rounds about 2 to 2½ inches in diameter, spacing them slightly apart. Brush the tops with the egg wash.
8. Bake at 425°F for 12–15 minutes until puffed, then reduce the heat to 375°F (190°C) and bake another 18–20 minutes until golden brown and dry to the touch.
9. Reduce heat to 300°F (149°C) and bake 15 minutes more to dry out the insides. Break one open to check — it should not be wet or eggy inside.
10. Transfer to a cooling rack and poke each puff with a toothpick or the tip of a paring knife to release steam. Cool completely.

To assemble:

1. Slice the tops off the cooled puffs about one-third down, using a serrated knife.
2. Whip the pâté with a stand or hand mixer until light and fluffy. Transfer to a piping bag and fill each puff to about 60% full (approximately 2 tablespoons).
3. Top each puff with 1 tablespoon of the cooled caramelized onions.
4. Add 1/8 teaspoon of raspberry preserves, then cap with the puff top. Serve immediately.