There are two critical steps in prepping kitchen walls.
Clean and degloss
Cleaning is critical: TSP or TSP substitute.
Deglossing: either light sanding (120 grit) or a chemical deglosser
Once a surface is deglossed, it should be primed, especially if going from one paint type to another.
A deglossed oil undercoat can be primed with a latex primer. Either oil or latex can then be used as a topcoat.
A deglossed oil undercoat can be primed with a oil primer. Either oil or latex can then be used as a topcoat.
It is better to stick within families of paint types, if you can, but going back and forth between types is possible with primers.
At this stage in coating, I'd let everything dry for a week, then sand lightly. If the previous layers are adhering, then proceed with latex primer, latex topcoat.
If,, however, they are sloughing or chipping or you can easily pull a layer off if you leave painters tape on overnight (and then pull it off), then you must sand down to the last well adhered layer and then repaint. Good luck.
On the face of it, that's not a bad idea.
I would only be worried about a few things, though:
One, is about the actual sheen that each brand presents. You really want to try both out on a scrap or hidden wall and make sure the brand you want to have show up on top is carefully selected.
Two, is about how easy will these varying coats be, for patch repairs as life happens.
Three, you generally don't double-coat your cut-ins. For example corners, around trimwork, etc., are mostly just a single coat (and you would roll as close to edges as possible), and doing two separate brands' coats on the entire wall surface will take a lot of (possibly thankless) work.
Best Answer
Technically yes as long as it is the same base (assuming your existing valspar paint was latex, you would need another latex paint).
The sheen or finish of your paint (flat/satin/semi/gloss) is of no concern and shouldn't provide a problem.
However, doing so likely won't do what you want. Assuming your dissatisfaction in texture is more than simply the difference between flat/gloss paint, it means the surface you painted was likely the source of your roughness. You need to smooth (sand) the surface before painting.