Preheat the broiler.
Cut the bagels in half horizontally and place under the broiler to toast for about 2 minutes, or until the top is browned.
To make the hollandaise sauce, mix the two egg yolks, lemon juice and sauerkraut juice together with an electric or manual whisk in a stainless steel bowl until the mixture starts to thicken and almost doubles in volume, about a minute.
Place the bowl over a saucepan containing simmering water (or in a double boiler on low heat). Continue to whisk rapidly, being careful not to let the eggs get too hot or they will scramble. Slowly pour in the melted butter and continue to whisk until the sauce is thickened. Add a bit of salt and fresh pepper and whisk until incorporated. Cover and put in a warm place until ready to use.
In a nonstick pan (preferably ceramic) on high heat, melt the coconut oil and add the four eggs. Fry until the whites are cooked and slightly brown at the edges but the yolks are still runny.
On a serving plate, place the toasted bagel halves, divide the salmon or lox on top of them, then place a fried egg on each half.
Drizzle some hollandaise sauce over each fried egg.
Serve immediately.