How bore out a shallow sink into a 2x10"?

I have a set of standing shelves that are built from standard home store 2×10"s. I wanted to repurpose the shelf as a place for my houseplants. When I water the plants, there is a lot of drainage water that seeps through the pot’s holes. I am trying to devise a way to drill a hole in the middle of each shelf and then bore/hollow out a very shallow decline in each of the shelves, so the water drains like a sink (almost like this: https://cdn.homedit.com/wp-content/uploads/2016/09/JM-Lifestyle-wood-wash-basin.jpg).

I don’t have a lathe, so I was thinking I could do this with an angle grinder (or perhaps a dremel tool?). Does anyone have suggestions on how best to accomplish this? I only have hand tools and am nervous about making the sink uneven.