Bring a large pot of water to a rolling boil. Add a significant amount of kosher salt. Cook 6 oz spaghetti according to package instructions until al dente. Reserve at least 1/4 cup of pasta water.
Meanwhile, heat 1 tbsp olive oil and 1 tbsp butter in a large saucepan. Add 2 anchovy filets and use a spatula to saute the anchovies while also breaking them up. Continue this until the anchovies are completely dissolved.
To this, add the zest of half a lemon and stir to combine until fragrant. Pour in 1/4 cup heavy cream and stir until fully combined. Simmer until slightly thickened.
Add the cooked and drained spaghetti and toss to combine. Add the pasta water in small increments until the sauce fully coats the pasta to your liking. Remove from the heat and squeeze the juice of 2 lemons into the pasta and stir to combine. Taste and adjust as needed.
In a small saucepan, heat 2 tbsp olive oil over medium heat and add 1 clove grated garlic. Let the garlic sizzle briefly before adding 1/4 cup of panko breadcrumbs. Stir and toast until the breadcrumbs are golden brown, then remove from the heat.
Serve the pasta hot and top with the garlic breadcrumbs, then garnish with thinly sliced chives.