To prepare the caramelized onion, melt the butter in a medium saucepan over medium-low heat. Add the olive oil and onion, season lightly with salt and pepper and stir to combine.
Cook for 20 – 30 minutes, or more, stirring frequently—it is okay if the onions brown lightly, which gives them color and sweetness, but don’t burn them. When done, the onions will have reduced to about 1/4 cup and be a pale golden color. Transfer to a small bowl and set aside.
To prepare the seasoned walnut spread, wrap the carrot, celery, shallot and bay leaf in a double thickness of cheesecloth and tie the bundle securely with twine. Put in a medium saucepan, pour in the milk and season with 1/4 teaspoon salt and 1/8 teaspoon pepper. Bring to a simmer over medium-high heat. Add the walnuts, reduce the heat, and poach in the simmering milk for 18 – 20 minutes, stirring once or twice.
Strain, reserving the milk. (Discard the cheesecloth bundle.) While the walnuts are still warm, put them in a food processor. Add 1/3 cup of the reserved milk and puree. Add additional milk by tablespoons, until the mixture is smooth and spreadable. Season to taste with salt and pepper, transfer to a bowl and set aside. (Any remaining milk may be used in a soup or sauce.)
To prepare the crostini, brush the baguette slices lightly with olive oil, put them in a single layer on a baking sheet, and place in a preheated 350ºF oven for about 5 minutes, to dry and crisp. Remove from the oven and gently rub each slice with garlic.
Spread about 1 tablespoon of seasoned walnut mixture on each crostini. In a small bowl toss the arugula with 1 tablespoon olive oil, then season lightly with salt and pepper. Drape a few leaves of arugula over each crostini and top with about 1/2 teaspoon caramelized onions.