In a saucepan, heat 1/4 cup of the oil over medium heat. Add the garlic and cook, stirring, until fragrant and soft, about 1 minute. Add the remaining 3/4 cup oil and the butter and cook gently, stirring, to melt the butter. Add anchovies, lower the heat to low, and cook, stirring, until the anchovies dissolve, 5 to 10 minutes. Remove from the heat and stir in the lemon juice. Adjust the seasoning to taste with salt, pepper, and cayenne (keeping in mind that the anchovies are salty).
Decoratively arrange the vegetables on a large platter and season lightly with salt. Transfer the anchovy mixture to an earthenware pot, or fondue pot fitted with a flame underneath, and place in the middle of the vegetables. Keep the sauce hot over a small flame. Serve immediately with long handled forks for dipping the vegetables into the sauce.